Dynamics of shear homeomorphisms of
tori and the Bestvina-Handel algorithm
Tali Pinsky and Bronislaw Wajnryb
Abstract
Sharkovskii proved that the existence of a periodic orbit of period
which is not a power of 2 in a one-dimensional dynamical system implies
existence of infinitely many periodic orbits. We obtain an analog of
Sharkovskii’s theorem for periodic orbits of shear homeomorphisms of
the torus. This is done by obtaining a dynamical order relation on the
set of simple orbits and simple pairs. We then use this order relation
for a global analysis of a quantum chaotic physical system called the
kicked accelerated particle.
1. Introduction
Given a dynamical system (X, f), a key question is which periodic
orbits exist for this system. Since periodic orbits are in general difficult
to compute, we would like to have the means to deduce their existence
without having to actually compute them.
Sharkovskii addressed the dynamics of continuous maps on the real
line. He defined an order C on the natural numbers, Sharkovskii’s or-
der (see [14]), and proved that the existence of a periodic orbit of a
certain period p implies the existence of an orbit of any period q C p.
We say the q orbit is forced by the p orbit. This offers the means of
showing the existence of many orbits if one can find a single orbit of
“large” period. For a dynamical system depending on a single parame-
ter, if periodic orbits appear when we change the parameter, they must
appear according to the Sharkovskii’s order. Hence, Sharkovskii’s the-
orem gives the global structure of the appearance of periodic orbits for
one dimensional systems. Ever since the eighties there has been inter-
est in obtaining analogs for Sharkovskii’s theorem for two dimensional
systems (see [4] and [16]).
A homeomorphism of a torus is said here to be of shear type if it
is isotopic to one Dehn twist along a single closed curve. Let h be a
shear homeomorphism, and let x be a periodic orbit of h. We can then
define the rotation number of x, see discussion in Section 2. Thus, a
rational number in the unit interval [0, 1) is associated to each orbit.
We consider orbits up to conjugation: orbits (x, f) and (y, g) are
similar (of the same type) if there exists a homeomorphism h of the
torus T 2 such that h is isotopic to the identity, h takes orbit x onto
orbit y and hfh−1 is isotopic to g rel y. We define below a specific
family of periodic orbits we call simple orbits. In this family there is a
unique element up to similarity corresponding to each rotation number;
hence they can be specified by their rotation numbers. We emphasize
it is not true in general that an orbit of a shear homeomorphism is
characterized by its rotation number.
Simple orbits are analyzed in Section 2. As it turns out (see Remark
in section 2), one simple orbit is indeed simple and does not force the
existence of any other orbit. More generally a periodic orbit is of twist
type if it does not force the existence of any orbit of different type with
the same rotation number. It is tempting to conjecture that the simple
orbits are the only orbits of twist type, but Lemma 2.4 shows that this
is false. We give there an example of an orbit of twist type which is
not simple. This example also shows that a periodic orbit with a given
rotation number does not necessarily force a simple orbit of the same
rotation number.
We turn in Section 3 to analyze pairs of orbits. Two coexisting simple
periodic orbits can form a simple pair and these are considered. The
pairs do force some more interesting dynamics, as follows. We denote
the integers by letters p, q and the rational numbers by r, s, t possibly
with indices. For a pair of simple orbits of rotation numbers q1
and q2
to constitute a simple pair, it is necessary that the rotation numbers
be Farey neighbors, i.e. |p2q1 − p1q2| = 1. We denote such a pair of
rational numbers by q1
We now define an order relation on the following set P of rational
numbers and pairs in the unit interval,
P = {r|r ∈ Q ∩ [0, 1)} ∪ {r ∨ s|r, s ∈ Q ∩ [0, 1)}.
Define the order relation on P to be
r ∨ s < t⇔ t ∈ [r, s]
r1 ∨ r2 < s1 ∨ s2 ⇔ s1, s2 ∈ [r1, r2].
where we denote by [r1, r2] the interval between r1 and r2, regardless
of their order.
Theorem 1.1. The order relation < on P describes the dynamical
forcing of simple periodic orbits. Namely, the existence of a simple pair
of periodic orbits with rotation numbers r∨s in P implies the existence
of all simple orbits and simple orbit pairs of rotation numbers smaller
than r ∨ s according to this order relation .
This is the main result of this paper and the proof is completed in
Section 4. The idea of the proof is as follows. Consider the torus punc-
tured on one or more periodic orbits of a homeomorphism h. Then h
induces an action on this punctured torus, and on its (free) fundamen-
tal group. Now apply the Bestvina-Handel algorithm to this dynamical
system. The idea of using the Bestvina-Handel algorithm was used by
Boyland in [6] and he describes the general approach in [5]. In our
case, after puncturing out a simple pair of orbits, applying the algo-
rithm yields an isotopic homeomorphism which is pseudo-Anosov. The
algorithm also offers a Markov partition for this system and we use
the resulting symbolic representation to find that there are periodic
orbits of each rotation number between the pair of numbers we started
with. Then we directly analyze the structure of the pseudo-Anosov
representative to show that all these orbits are in fact simple orbits.
Furthermore any two of them corresponding to rotation numbers which
are Farey neighbors form a simple pair. Finally we establish the iso-
topy stability of these orbits using results of Asimov and Franks [2] and
Hall [11]. Thus, the orbits exist for any homeomorphism for which the
simple pair exists, and are forced by it.
One should compare this result with a very strong theorem of Doeff
(see Theorem 3.6), where existence of two periodic orbits of different
periods for a given shear homeomorphism h implies existence of peri-
odic orbit of every intermediate rotation number. However an explicit
description of these orbits is not given, while our stronger assumptions
imply existence of simple periodic orbits and simple pairs of orbits. It
may be true that the existence of any two periodic orbits with differ-
ent rotation numbers implies the existence of a simple orbit with any
given intermediate rotation number, but we feel that the evidence is not
strong enough to make a conjecture either way, in particular in view
of Lemma 2.4. Even more difficult question is to determine whether
there exists a simple pair of periodic orbits in the situation of Doeff
Theorem. First one should try to find a pair of simple orbits which is
not a simple pair while the rotation numbers are Farey neighbors, but
it is very difficult to understand the geometry of the pseudo-Anosov
homeomorphism which arises in this situation.
This research was originally motivated by a question we were asked
by Professor Shmuel Fishman: Is there a topological explanation for
the structure of appearance of accelerator modes in the kicked particle
system. In section 5 we give a description of the kicked particle system.
This system turns out to be described precisely by a family of shear
homeomorphisms of the torus. The existence of accelerator modes is
equivalent to existence of periodic orbits. The global structure of this
system is given by the order relation in Theorem 1.1, while it cannot
be directly computed due to the complexity of the system.
The authors would like to thank Professor Shmuel Fishman for offer-
ing valuable insights, Professor Italo Guannieri for some critical advice,
and Professor Philip Boyland for many indispensable conversations.
2. Simple orbits
Let {x1, ....., xN} be a set of points belonging to one or more pe-
riodic orbits for a homeomorphism f of a surface S. The dynamical
properties of this set of orbits are captured by the induced action of f
on the complement S0 = S \ {x1, ...., xN} in a sense that will shortly
become clear. Choose any graph G which is a deformation retract of
the punctured surface S0. A homeomorphism of S0 then induces a map
on G. The converse is also true: a given map of G determines a homeo-
morphism of S0 up to isotopy. Therefore we specify the periodic orbits
we analyze in terms of the action on a graph which is a deformation
retract of the surface after puncturing out the orbit.
Denote by GN the graph obtained by attaching N small loops to the
standard unit circle at the points exp(2πj
i), j = 0, . . . , N − 1.
Definition 2.1. We call a periodic orbit x = {x1, . . . , xN} for a shear
homeomorphism f on the two-torus a simple orbit if the following hold.
(1) There can be found a graph G which is a deformation retract
of T0 = T
2 \ x as on Figure 1 such that G is homeomorphic to
We call the loop in G corresponding to the unit circle the hor-
izontal loop and the loops attached to it the vertical loops.
(2) There exists a homeomorphism f̃ of T0 isotopic to f rel x (i.e.
the isotopy is fixed on x) so that a neighborhood of G is invari-
ant under f̃ and the induced action on G satisfies: (a) There
exists a fixed number k ∈ {0, .., N − 1} such that each vertical
loop is mapped k loops forwards (clockwise along the unit cir-
cle) to another vertical loop. (b) The horizontal loop is mapped
to itself with one twist around one of the vertical loops.
Figure 1. A standard graph for a simple orbit
Figure 2. The action on a standard graph for a simple orbit
Remark. A homeomorphism h for which we are given a simple periodic
orbit must be of shear type as we can deduce from the action on the
homology of the non-punctured torus.
For a shear homeomorphism there exists a basis for the first homol-
ogy for which the induced map is represented by the matrix
From here on we refer to any two axes given by an homology basis that
gives us the above representation as standard axes. The horizontal loop
and one of the vertical loops in a graph for a simple orbits constitute
a standard basis.
Definition 2.2. Let h be a shear homeomorphism, and let ĥ be a lift
of h to the universal cover (a plane). For any periodic point x of h
of period p, ĥp maps any lift x̂ of x the same integer number q along
the horizontal axis away from x̂, in a standard choice of axis (and x̂ is
possibly mapped some integer number along the vertical axis as well).
We can then define the rotation number of x to be ρ(x) = q
mod 1.
The rotation number does not depend on the lift ĥ of h.
(The rotation number is often define relative to the given lifting of
the homeomorphism h and is not computed modulo 1, but we want it
to depend only on the orbit and not on the lifting. In particular we
want a simple orbit to have well defined rotation number independent
of the lifting of h.)
Remark. In the case of a homeomorphism isotopic to a Dehn twist on a
torus, which is our interest here, it can be easily shown that the abelian
Nielsen type equals exactly the rotation number defined above.
There exists a simple orbit for any given rational rotation number
r ∈ [0, 1), and it is unique up to similarity. Denote the similarity class
by r̂. In the following we use the word vertical to describe the y axis,
in a standard choice of axis for f (the direction along which the twist
is made).
Lemma 2.3. Let x be a periodic orbit for a shear-type homeomorphism
f of T2 for which there exists a family of vertical loops such that they
bound a set of annuli each containing one point of the periodic orbit,
and this family is invariant under a homeomorphism f̃ isotopic to f
rel x. Then x is a simple orbit.
Proof. Choose a vertical loop l of the invariant family. f is orientation
preserving, and so is f̃ . The first loop to the right of l is therefore
mapped to the first loop to the right of f̃(l). Hence, the vertical loops
in the invariant family are all mapped the same number of loops to the
right.
Now we have to find a horizontal line with the desired image. We
write the invariant family of loops as {li}
i=1, where p = period(x),
ordered along the horizontal axis. We choose another family of vertical
loops {mi}
i=1, such that mi is contained in the annulus between li and
li+1 (lp+1 = l1), and passes through the periodic point xi also contained
in this annulus. Choose a point a1 6= x1 on m1. f can be adjusted in
such a way that the new homeomorphism f̃ leaves both families of
vertical loops invariant, and in addition, so that a1 be a periodic point
of f̃ with period p. We denote the orbit of a1 by {ai}
i=1 where ai ∈ mi
for 1 ≤ i ≤ p.
Choose a line segment n1 connecting a1 to a2, so it crosses the annulus
between m1 and m2 from side to side. We choose nj+1 to be the line
segment f̃ j(n1) for 1 ≤ j ≤ p − 2. The boundary points of nj and nk
coincide whenever they lie on the same vertical loop.
Now, we look at the horizontal loop n =
j=1 nj. Each segment of n
is mapped exactly to the next segment, except np−1 which is mapped
into the annulus between m1 and m2. Since the mapping class group
of an annulus is generated by a twist with respect to any loop going
once around the annulus we may assume, that np is mapped to n1 plus
a number of twists along such a loop. On the other hand we know that
f(n) is homotopic to itself plus one twist in the negative direction (on
the closed torus), so f̃ maps n to itself plus one negative twist along
this loop. By further adjustment of f̃ we may assume the twist is made
along l2. Thus the union of the vertical family {li} with n chosen as
above constitute a graph showing x to be a simple orbit. �
The Thurston-Nielsen classification theorem, see [7], states that any
homeomorphism f on a closed connected oriented surface of negative
Euler characteristic is isotopic to a homeomorphism f̃ which is
(1) pseudo Anosov, or
(2) of finite order, or
(3) reducible.
where a homeomorphism φ is called of finite order if there exists a
natural number n such that φn = id. A homeomorphism φ is called
pseudo-Anosov if there exists a real number λ > 0 and a pair of
transverse measured foliations (Fu, µu) and (F s, µs) with φ(Fu, µu) =
(Fu, λµu) and φ(F s, µs) = (F s, 1
µs). A homeomorphism φ on a sur-
face M is called reducible if there exists a collection of pairwise disjoint
simple closed curves Γ = {Γ1, ...,Γk} in int(M) such that φ(Γ) = Γ
and each component of M \Γ has a negative Euler characteristic. The
representative f̃ in the isotopy class of f which is of one of the three
forms above is called the Thurston-Nielsen canonical form of f .
When the surface has a finite number of punctures and φ permutes
the punctures then the same is true except that in the case of pseudo-
Anosov map we treat the punctures as distinguished points (there is a
unique way to extend a homeomorphism to the distinguished points)
and we allow an additional type of singularities of the measured foli-
ations, the 1-prong singularities at the distingushed points (See [11],
and section 0.2 of [3]).
Of course homeomorphism f is reducible with respect to a simple
orbit since it contains an invariant family of loops {li} and the comple-
ment of the invariant family consists of punctured annuli (which have
negative Euler characteristic).
Remark. A homeomorphism f with a simple orbit x can be constructed
in such a way that x is the only periodic orbit of f . The invariant
set of vertical loops is evenly spaced with the distance between the
consecutive loops equal to 1/p. The loops are moved by q/p to the
right and by fixed irrational number downward. Punctures (the points
of the periodic orbit) are also evenly spaced and have the same height.
The vertical lines containing punctures are moved by q/p to the right.
The punctures keep their height and all other points of the loop move a
little downwards. Every other vertical line is moved to another vertical
line by a little more than q/p to the right (not all lines by the same
distance).
Example 1. Not every periodic orbit for a shear homeomorphism is
reducible. Consider the homeomorphism h described on Figure 3. It
takes the graph on the left of Figure 3 to the graph on the right and is
a shear homeomorphism. It has a periodic orbit of order 2, shown on
the pictures, with rotation number 1/2 and it is pseudo-Anosov in the
complement of the orbit.
Figure 3.
Lemma 2.4. There exists an orbit of twist type for a shear homeomor-
phism of the torus which is not simple.
Proof. We construct an example of an orbit of length 4 with the rota-
tion number 1/2. It cannot be a simple orbit and yet we prove it does
not force the existence of any periodic orbit not similar to itself, and is
thus of twist type. Such examples may be known, possibly considered
for a different phenomena. We include it here in order to show the
independence of our results.
We represent the torus as the unit square with the opposite sides
identified. The points A1, A2, A3, A4 of the orbit are spaced evenly on
the horizontal middle line with the x-coordinate 1/8, 3/8, 5/8, 7/8.
We split the square into 2 equal parts U1 and U2 by the vertical line
x = 1/2. Homeomorphism h translates U1 to the right to U2. Vertical
lines go to vertical lines, lines x = 0 and x = 1/2 move downward by
an irrational number α < 1/40 and the movement is damped out to the
horizontal translation for t < α and t > 1/2− α, so the other vertical
1 2 3 4 5 6 7
Figure 4. Some vertical lines in U2
� �' $
'' %%%�
' %' %��
%' %' %�
1 2 3 4 5 6 7
Figure 5. The foliation in U2 realizing a non-simple
orbit of twist type.
lines are translated horizontally by 1/2. In particular A1 moves to A3
and A2 moves to A4.
The restriction of h to U2 is defined in two steps. The second step
simply translates U2 horizontally by 1/2 to the right (which is the
same as the translation by 1/2 to the left). The first step is isotopic
to the half-twist along the segment connecting A3 and A4, followed by
the Dehn twist with respect to the right side (right boundary of the
cylinder). In particular it switches A3 and A4. We shall prove that we
can construct such h for which h2 has no fixed points and therefore h
has no periodic orbit of length 2 and in particular no simple orbit with
the rotation number 1/2.
We describe the first step of h restricted to U2. Figure 4 shows some
vertical lines in U2, the big dots show the points A3 and A4 of the
periodic orbit. Figure 5 shows their images under the first step. In
these pictures U2 is represented as a square, to make more space, but
in the reality the base has length 1/2 and the height is equal to 1. Line
x = 1/2 is mapped to itself and moves downward by α. The near by
vertical lines (for t < 1/2+1/30) are moved to the vertical lines and to
the right where line x = 1/2+1/30 is moved to the line x = 1/2+1/20.
For t ∈ (1/2 + 1/30, 1−1/20) the line x = t is moved to a curve Lt and
for t ∈ (1−1/20, 1) the line x = t moves to a vertical line to the right of
it and downward, to get the full Dehn twist plus a movement downward
by α when we get to the line x = 1. For t ∈ (1/2+1/30, 1−1/20) curve
Lt starts at a point on the top side to the right of x = t, it moves to the
left, then to the right, then to the left again and ends at the bottom
side (exactly below its starting point). In particular each vertical line
meets Lt in at most two points. Some lines Lt are shown on Figure 5.
We may arrange it in such a way that there exist t0, t1 such that
1/2 < t0 < t1 < 1 and the line x = t:
is disjoint from Lt, and lies on the left side of Lt when t < t0;
meets Lt at one point for t = t0;
meets Lt at two points when t0 < t < t1;
meets Lt at one point for t = t1;
is disjoint from Lt and lies on the left side of Lt when t1 < t < 1.
We get a new trivial foliation of the annulus U2. In step 1 we map
the vertical foliation onto the new foliation Lt. We can further change
the first step moving each leave Lt along itself to reach the following
goal. Let Pt, Qt denote the intersection points of x = t with Lt, Pt
lies below Qt (the points coincide for t0 and t1). For t = t0 the line
x = t meets Lt in one point Pt. We may assume that the image of Pt
in Lt lies in the part below Pt. Then for the nearby leave the images
of both points Pt and Qt lie in the lower part of Lt below the point
Pt (see the small dots on the first curve in Figure 5). The images of
Pt and Qt lie further away from each other when we move to the right
(see the small dots on the second curve). The third line passes through
A3, its image Lt passes through A4 and the images of Pt and Qt lie on
different sides of A4 along the third curve. Next the upper point Qt
moves backwards along Lt and when we reach the fourth line of Figure
4 (also shown on Figure 5) it coincides with the point Pt on Lt. Next
the image of Qt lies inside the arc of Lt between the points Pt and Qt
and when we reach the fifth line on Figure 4, which passes through the
point A4, then the curve Lt passes through A3 and the image of Qt on
Lt lies above A3 (see Figure 5). When we move further to the right
the image of the point Qt moves again forward towards the image of Pt
and at the line number 6 on Figure 4 the image of Qt again coincides
with Pt at the intersection of x = t with Lt. Next the images of Pt and
Qt move further down and gets close together and when t = t1 we have
one intersection point Pt and its image lie below Pt along Lt. Step 1
has no fixed points. Step 2 translates U2 to U1.
We now consider the homeomorphism h2. We start with U1. Any
point on x = 0 and x = 1/2 moves down by 2α. Any point with
x ∈ [0, 1/30] moves to a point with a bigger x-coordinate. Any point
with x ∈ [1/30, 9/20] moves horizontaly by 1/2 then we apply step 1,
which has no fixed points, and then the point moves again horizontaly
by 1/2 so it comes to a new point. Any point with x ∈ [9/20, 1/2]
moves to a point with a bigger x-coordinate.
For points in U2 the situation is similar. Any point with x ∈ (1/2, 16/30]
moves to a point with a bigger x-coordinate. Any point with x ∈
[16/30, 19/20] moves under the first step to a new point with the x-
coordinate in [11/20, 29/30] and then moves horizontaly twice by 1/2.
Finally any point with x ∈ [19/20, 1] moves to a point with a bigger
x-coordinate. Homeomorphism h2 has no fixed points and h has no
periodic points of order 2.
We now show that there exists a homeomorphism f isotopic to h
in the complement of the orbit A1, A2, A3, A4, which has only periodic
orbits similar to this orbit and periodic orbits of order 2. We consider
parts U1 and U2 as before. The restriction of f to U1 translates it hor-
izontaly by 1/2. In U2 we choose two circles with center (3/4, 1/2) and
radius 1/7 and 1/6 respectively. We rotate the interior of the smaller
circle by 180 degrees. The rotation is damped out to the identity at
the outer circle and the intermediate circles are moving out towards
the outer circle. The exterior of the outer circle with x < 19/20 is
pointwise fixed. The lines with x > 19/20 move to the right and down
to get the full Dehn twist when we get to the line x = 1. The second
step of f restricted to U2 translates it horizontally by 1/2. Now each
point inside the smaller circle, different from its center (which has pe-
riod 2), belongs to an orbit similar to A1, A2, A3, A4. Points between
the circles and points with x ∈ (19/20, 1) are not periodic and other
points in U2 have period 2 and the same is true for the corresponding
points in U1.
Therefore the orbit A1, A2, A3, A4 does not force any periodic orbit
not similar to itself. �
3. Simple orbit pairs
Let x and y be two coexisting simple periodic orbits, for a homeo-
morphism f of T2 (f must be of shear type), with rotation numbers q1
and q2
respectively. Assume p1 > p2, i.e., y has lesser period than x.
Definition 3.1. We call the pair of orbits a s imple pair if
• We can find an embedded graph G in their complement home-
omorphic to Gp1 as on Figure 6.
Figure 6.
Each component in the complement of the graph is a topolog-
ical rectangle which contains exactly one point of orbit x and
at most one point of orbit y.
• The homeomorphism f acts on this graph in the following way:
each vertical loop except one moves to another vertical loop,
there is one vertical loop denoted l such that f(l) is a vertical
loop m plus a small loop around one point of the (shorter) y
orbit, in a rectangle adjacent to line m on the right (as on Figure
7) or on the left, and the horizontal line is mapped to itself plus
a twist in the negative direction around f(l), as on Figure 7.
The graph which appears in Definition 3.1 divides the torus T2 into p1
rectangles. The homeomorphism f moves each vertical loop the same
Figure 7.
distance, say k rectangles, to the right except for the small additional
loop for line l. Let R0 be the rectangle adjacent to m in which the
small loop in the image of the graph occurs. R0 must contain exactly
one point of each orbit. We denote these points x0 and y0 respectively.
Under p1 iterations of f the point x0 runs q1 times around the whole
torus, that is q1p1 rectangles to the right. So, p1k = q1p1 and k = q1.
The point y0 is mapped to itself after p2 iterations.Under each itera-
tion the image of y0 is mapped k (= q1) rectangles to the right, except
the last iteration under which it is moved an additional rectangle to the
right or left. Altogether it has moved p2q1± 1 rectangles. At the same
time, it is mapped around the torus q2 times, hence q2p1 rectangles.
This means p2q1 ± 1 = q2p1. Thus for a simple pair of periodic orbits
the rotation numbers q1
and q2
are Farey neighbors and the additional
loop is on the right (as on Figure 7) if and only if q2
. Denote
by r̂ ∨ ŝ the similarity class of a simple pair corresponding to a pair of
Farey neighbors r ∨ s.
Consider again the points x0 and y0 in the rectangle R0. Continue
the notation to all the points of x and y by xi = f
i(x0) and yi = f
i(y0).
We draw a small loop around each of the points of y. The union of
these loops will be the peripheral subgraph P for the Bestvina-Handel
algorithm, since we may assume the union of these loops to be f -
invariant. Now we consider separately two cases. Case 1 will be the
case in which m is the left boundary curve of R0, while in case 2 it is
the right boundary (in other words in the first case q1
and in the
second case q1
). Choose some point on the loop around y0 and
connect it, by a curve l0, to a point on the section of the horizontal line
in R0, in case 1 from below the segment and in case 2 from above.
Figure 8.
Then f(l0) is a curve connecting the loop around y1 and the corre-
sponding horizontal segment. We denote it by l1, and do the same for
each yi. After adding the above edges to the graph case 1 is topologi-
cally as in Figure 8.
The inclusion G ↪→ S0 is a homotopy equivalence (where S0 is the
punctured torus). We know the action of f on all edges of G except
for the curve lq1−1 connecting yq1−1 and the horizontal segment in the
corresponding rectangle. It’s image is a curve connecting the horizontal
segment in the rectangle adjacent to m which is not R0 to the loop
around y0. This image might wind around a disk containing y0 and
xq1−1 as in Figure 9
Figure 9.
The graph and its image for case 2 are exactly the same except that
the loops are connected to the horizontal segments from above. We
shall prove in Proposition 3.2 that we may assume that the image of
the segment lq1−1 has no winding. Hence we draw from now on the
graph images without winding, and we may assume the graphs given
in Figure 10 also have an invariant neighborhood by a further isotopy
of f .
Figure 10. The standard graph for a simple pair, case
1 on the left and case 2 on the right.
The action of f (up to isotopy) on this graph is given by one of the
actions on Figure 11, drawn in some regular neighborhood of the graph,
where each vertical loop moves q1 loops to the right.
Figure 11. The action on a standard graph for a simple
pair, for both cases respectively.
Proposition 3.2. Let {x, y} be a simple pair with the graph as on
Figure 8 and with windings as on Figure 9. Then there may be chosen a
different invariant graph, which also makes {x, y} a simple pair, whose
image is without winding.
Proof. To simplify the picture we prove the proposition for rotation
numbers 1
and 1
. The general proof proceeds in the same way. We
start by looking at a simple pair {x, y} of rotation numbers 1
and 1
a homeomorphism f with the corresponding invariant graph G given
so that the action on it is without any twists, as on the left side of Fig-
ures 10 and 11. We now choose a different system of curves (a different
graph), in a small neighborhood of G as in Figure 12, which will serve
as a new graph for the pair.
Figure 12.
Solid lines are the new vertical loops and dashed lines are the new di-
agonal segments like in figure 10. The horizontal loop consists of the
dashed lines and the long pieces of the solid lines. To move from left
to right along the horizontal line, move along a dashed line and turn
to the left when meeting a solid line. Continue up along a vertical loop
and then along the next dashed line. We add to this graph the periph-
eral subgraph and the connecting segments and get the graph H as on
Figure 13. It is clear that topologically the graph H has the same form
as the graph on Figure 10 and that it has an invariant neighborhood.
The reader can check (using the precise knowledge of the image of
each edge of the original graph) that the action of f on the graph H
has the properties required from a simple pair. Each vertical loop is
mapped onto another vertical loop except for one loop l for which f(l)
is equal to a loop m plus a loop around the next periodic point yp
of the shorter orbit. The horizontal loop is mapped onto itself plus a
negative Dehn twist along f(l). Consider the image s of the segment
which connects the horizontal loop to the periodic point yp−1. When
the action has no twist then s moves along the horizontal loop in its
Figure 13.
positive direction until it meets the original segment connecting to yp
and then it follows along the segment. However in our case s goes first
backwards along the horizontal loop than moves in the counterclockwise
direction along the boundary of the ”rectangle” adjacent to the vertical
loop m and finally follows the horizontal loop and the segment to yp.
This means that the action f on the graph H has one positive twist.
We proved that a simple pair for a shear homeomorphism with a
given graph and a given action without twists can be given another
graph which also describes it as a simple pair and the action on the new
graph has one positive twist. This process is reversible. Therefore, by
induction, we can add or remove any number of twists using a suitable
graph. This implies Proposition 3.2. �
Hence for a simple pair the action on a spine is given by Figures 10
and 11. We can now apply the Bestvina-Handel algorithm (see [3]),
endowing a neighborhood of G with a fibered structure in the natural
way. The algorithm specifies a finite number of steps which we apply
to the graph G, altering G together with the induced action on it, but
without changing the isotopy class of f on T2 \ (x ∪ y). When the
algorithm terminates, it gives a new homeomorphism f̃ which is the
Thurston Nielsen canonical form of f .
For simple pairs, the action in each of the two cases above is easily
seen to be tight, as no edge backtracks and for every vertex there are
two edges whose images emanate in different directions. The action
has no invariant non-trivial forest or nontrivial invariant subgraph and
the graphs have no valence 1 or 2 vertices. This is the definition in [3]
for an irreducible map on a graph.
Definition 3.3. Assuming g, the induced map on the graph itself, does
not collapse any edges, there is an induced map Dg, the derivative of
g, defined on
{(v, e)|v is a vertex of G, e is an oriented edge emanating from v}
by Dg(v, a) = (g(v), b) where b is the first edge in the edge path g(a)
which emanates from g(v).
Definition 3.4. We say two elements (v, a) and (v, b) in L corresponding
to the same vertex v are equivalent if they are mapped to the same
element under D(gn) for some natural n. The equivalence classes are
called gates
The gates in each of the cases above are given by Figure 14, indicated
there by small arcs. There is no edge which g sends to an edge path
Figure 14.
which passes through one of the gates - enters the junction through one
arm of the gate and exits through the other. Such an irreducible map
is efficient. i.e., this is an end point of the algorithm. Now, since there
are edges mapped to an edge path longer than one edge, we arrive at
our next theorem.
Theorem 3.5. A homeomorphism f of the two torus for which a simple
pair of periodic orbits exists is isotopic to a pseudo-Anosov homeomor-
phism relative to this pair of orbits.
Let f be a shear type homeomorphism of the torus, and fix a lift f̃
of f . Define the lift rotation number of a point x ∈ T2 to be
ρ(x, f̃) = limn→∞
(f̃n(x̂)− x̂)1
for any lift x̂ of x, when the limit exists, where the subscript 1 denotes
the projection to the horizontal axis. Define the rotation set ρ(f̃) of f̃
to be the set of accumulation points of{
(f̃n(x̂)− x̂)1
|x̂ ∈ R2and n ∈ N
Then, the above theorem follows from the following much more gen-
eral theorem by Doeff, see [9] and [10].
Theorem 3.6. (Doeff) Let h be a shear type homeomorphism of T2,
and fix a lift h̃ of h. If h has two periodic points x and y with ρ(x, h̃) 6=
ρ(y, h̃) then h is pseudo-Anosov relative to x and y. Furthermore, the
closure of the rotation set is a compact interval, and any rational point
r in the interior of this interval corresponds to a periodic point x ∈ T2
with ρ(x, h̃) = r.
In particular, Doeff proves existence of two periodic orbits of dif-
ferent rotation numbers implies existence of an orbit for any rational
rotation number between these two. But he does not give any charac-
terization of these orbits. Example 1 shows two different orbits, both
with rotation number equal to 1/2, one of which is pseudo-Anosov,
and the other reducible. Thus the rotation number does not give much
information about the orbit and in this sense this theorem does not
give a satisfactory dynamical understanding of what is happening in
regions of coexistence of orbits. In contrast with Doeff’s general theo-
rem, we get results for a very specific family of periodic orbits, but for
this family we are able to give exactly the orbits forced by others, as
we show in section 4.
In our case the canonical form f̃ of f we get by applying the Bestvina-
Handel algorithm is a pseudo Anosov homeomorphism. When this is
the case, the algorithm gives a canonical way of endowing a regular
neighborhood S0 of G with a rectangle decomposition {R1, ..., RN}.
The decomposition is a Markov partition for the homeomorphism PSL.
A Markov partition for a dynamical system offers a symbolic repre-
sentation for the system in the following way. Let ΣN be the subset of
the full N -shift (the set of bi-infinite series on N symbols), where N is
the number of rectangles in the decomposition. Let Σ be a subset of
ΣN defined by
Σ = {s = (..., sn, sn+1, ....) : Rsn ∩ f̃−1Rsn+1 6= ∅}
On Σ we naturally define a dynamical system with the operator of the
right shift denoted by σ, and (Σ, σ) is called the subshift corresponding
to the dynamical system. Σ can be completely described by stating
which transitions k → m for k,m ∈ {1, ...., N} are allowed (i.e., for
which k,m, f̃−1Rm ∩ Rk 6= ∅). See [1] for the definitions and for a
proof that in this case we can define a map π : Σ→ S0 by
π : s 7→
n=0 f̃
nRs−n ∩ .... ∩ f̃−nRsn
which satisfies the following properties:
• πσ = f̃π,
• π is continuous,
• π is onto.
We take here the set of sequences with the Tichonoff topology. Thus
a periodic point in the symbolic dynamical system which is just a peri-
odic sequence corresponds to a periodic point in the original dynamical
system.
To obtain the Markov partition in our case as in [3], we thicken the
edges of the graph to rectangles. In particular, the rectangles can be
glued directly to each other without any junctions. This can be done
in a smooth way, endowing S0 with a compact metric space structure
by giving a length and width to each rectangle, consistently. Each edge
of the standard graph for the pair (figure 10) corresponds to one rec-
tangle, except the edge which is mapped to the loop around y0. This
edge we divide in two (this is necessary to avoid having a rectangle
intersecting twice an inverse image of another rectangle). Now we have
edges of 7 different types on the graph. The vertical loops of the graph
consist of long edges we denote as A edges, and short edges we call
B edges. The loops around the points of the y orbit and vertical seg-
ments connecting the loops to the diagonal edges we call C’s and D’s
respectively. In rectangles which contain two punctures and therefore
two diagonal edges we call the upper ones L edges and the lower ones
K edges in the first case, and the lower ones L edges, upper ones K
edges, in the second case. The last type of edges are diagonals of once
punctured rectangles, these we call M edges.
Next, we label the rectangles in order to have explicitly the transition
rules:
• For 0 ≤ i ≤ p2 − 1 denote the rectangle corresponding to the
D edge connecting the loop around yi to the diagonal by ri+1.
Denote the rectangle corresponding to the C edge which is the
loop around yi by rp2+i+1.
• For 1 ≤ i ≤ p2, denote the rectangles corresponding to the L
and K edges connected to ri by r2p2+i and r3p2+i respectively.
Figure 15.
• Denote the rectangle corresponding to the A edge belonging to
the vertical line we referred to as m by r4p2+1 and the B edge
which is part of the same line m as r4p2+p1+1.
• For the vertical line f i(m) denote it’s A and B rectangles by
r4p2+1+i and r4p2+p1+1+i respectively for all 1 ≤ i ≤ p1 − 2
• For the vertical line fp1−1(m), denote it’s A edge as by r4p2+p1 .
There are two rectangles corresponding to the B edge as ex-
plained above, denote the lower one by r4p2+2p1 and the upper
one by r4p2+2p1+1.
• Label the p1− p2 remaining rectangles corresponding to the M
edges by starting with the first of these to the right of m, and
then continuing by the order along the horizontal axis, denoting
them by r4p2+2p1+2, ... ,r3p1+3p2+1
Finally, we can look at the diagram in figure 16, showing the set of
rectangles and transitions in this Markov partition which we now use.
A periodic symbolic sequence of allowed transitions gives as ex-
plained a periodic point in the original dynamical system. Therefore
by this diagram we can easily find other periodic orbits on the torus
that must exist for f . We will later prove that these orbits are in
fact simple, but this will require some more work. Hence, by this
diagram we prove only existence of orbits with specified rotation num-
bers. For every pair (n,m) of natural numbers, n,m 6= 0, by starting
from rp1+4p2+1, going n times around the first loop in the diagram
{rp1+4p2+1, . . . , r2p1+4p2}, then going m-1 times around the second loop
{r1, . . . , rp2} (and skipping it if m=1) and then returning through the
final sequence {r2p2+1, . . . , r3p2} to rp1+4p2+1, we get a periodic symbolic
allowed sequence, and so a new periodic orbit we denote On,m. These
Figure 16. Some of the rectangles in the Markov par-
tition, where the arrows denote allowed transitions be-
tween them
symbolic sequences are all different and hence so are the periodic orbits.
We look at a point p ∈ On,m such that p is in the rectangle rp1+4p2+1.
For the first n ·p1 iterations of p, corresponding to each time the upper
loop in the diagram appears in the symbolic sequence of p, the images
are contained in the B edges. The vertical loops are mapped under f
retaining the same ”horizontal distance” from the periodic points from
the x orbit to their left. So, p is mapped a total distance of n · q1 along
the horizontal axis under fn·p1 .
Similarly, point fnp1(p), which lies in the rectangle r1 corresponding
to D edge, is mapped a distance q2 along the horizontal axis under each
iteration of fp2 , for every occurrence of the second loop in the symbolic
sequence of p. This is because the D edges retain their distance from
the y orbit points below them. The final sequence in the symbolic
representation of p until the return to the first loop also corresponds
to the horizontal distance q1. These last points of the periodic orbit lie
in rectangles corresponding to L edges. So p is mapped a horizontal
distance of nq1 + mq2 under f
np1+mp2 . Hence, the new orbit On,m has
rotation number nq1+mq2
np1+mp2
See [12] for a proof that any two Farey neighbors span this way all
rational numbers between them, that is all rationals between q1
and q2
are of the form nq1+mq2
np1+mp2
. So we found a periodic orbit of any rational
number between the two original rotation numbers q1
and q2
Note we have found these simple periodic orbits for the Thurston-
Nielsen canonical form of the homeomorphism f we started with. It
remains to relate these periodic orbits to the periodic orbits of f itself.
Recall the following definition from [2].
A periodic point x0 ∈ S of period p for homeomorphism f0 is called
unremovable if for each given homomorphism f1 with ft : f0 ' f1 there
is a periodic point x1 of period p for f1and an arc γ : [0, 1] → S with
γ(0) = x0, γ(1) = x1 and γ(t) is a periodic point of period p for ft.
It was proven by Asimov and Franks in [2] that every periodic orbit of
a pseudo-Anosov diffeomorphism is unremovable. Thus orbits found for
the pseudo-Anosov representative exist for any other homeomorphism
in its isotopy class. This yields all these periodic orbits exist for the
original homeomorphism f as well. Thus we get theorem 3.6 for our
specific case:
Theorem 3.7. If there exists a simple pair of orbits for a homeomor-
phism f of the torus of abelian Nielsen types s and t which are Farey
neighbors, there exists a periodic orbit for f with abelian Nielsen type
equal to r for every rational number r between s and t.
4. The order relation
For any simple pair q̂1
∨ q̂2
, The orbit O1,1 out of the family of
new orbits we constructed above has rotation number equal exactly to
q1+q2
p1+p2
. This orbit corresponds to the symbolic sequence rp1+4p2+1 →
rp1+4p2+2 → ....→ r2p1+4p2 → r2p2+1 → ....→ r3p2 → rp1+4p2+1 as in the
diagram in figure 16. So we have a list of rectangles, each containing
exactly one periodic point from the new orbit O1,1. We denote the point
of O1,1 that is in a rectangle rj by oj. Graphically, assuming the first
case map, when we draw the rectangle decomposition corresponding to
the standard graph as in figure 10 we get Figure 17.
We will now show that for any simple pair q̂1
∨ q̂2
the orbit O1,1
is a simple orbit, and forms a simple pair with each periodic orbit of
the pair, that is with q̂1
and q̂2
. For the first assertion, we define
Figure 17. The intermediate orbit denoted by black
circles. The gray areas are junction, which can be
deleted, and the rectangles can be glued directly to one
another.
a family of vertical loops as follows: we choose a vertical loop that
crosses both rectangles corresponding to the m line and passes to the
right of the periodic point op1+4p2+1. Denote this loop by A. It is shown
graphically in figure 18. All its images under f until the p1st iteration
are exactly of the same form, as the rectangles are simply mapped to
the right without changing their forms. Its p1st image is the first time
it returns to the same rectangles, and is determined by the images of
the corresponding vertical edges of the graph. These images are shown
in figure 11. We use the fact f preserves orientation to determine the
relation between the image of the curve and the points of O1,1. We
denote this image by B. It is shown in figure 18.
By similar considerations, knowing the rectangles containing B in the
original picture (Figures 10 and 11) the rectangle adjacent to fp1−1(m)
on the left contains a point of the y-orbit and therefore contains a
rectangle of type L of Markov partition. This rectangle contains the
point o3p2 of the new orbit. Line A lies to the right of m and of op1+4p2+1
therefore fp1−1(A) lies to the right of o3p2 and to the right of o2p1+4p2 .
It follows that the line B = fp1(A) lies to the right of op1+4p2+1 and to
the right of o2p2+1, as shown on Figure 18. Also B can be isotoped to
the right of A relative to the points of the new periodic orbit. Next
p2 − 1 iterations of f translate A and B and whole rectangle adjacent
to m on the right to the right. We arrive at the rectangle adjacent to
Figure 18. The intermediate orbit with the family of
vertical loops
l on the left containing point o3p2 of the new orbit. The point o2p1+4p2
lies in a rectangle on the line l. The loop fp2−1(B) lies to the right of
o3p2 and to the left of o2p1+4p2 therefore the loop C = f
p2(B) lies to the
right of op1+4p2+1 and to the left of o2p2+1, as shown on Figure 18. Also
p2 iterations of f take line m to a distance p2q1 = p1q2 − 1 rectangles
to the right, which means one rectangle to the left of line m. Since A
and B are to the right of the point op1+4p2+1 in line m and since this
point moves to the leftmost point in the new periodic orbit shown on
Figure 18, loop C must be to the right of it, as in Figure 18. The point
o2p1+1 may be above or below the loop C but this does not change the
discussion bellow.
Note that if we disregard the orbits x and y of the original pair we
can isotop C to A relative the points of the new orbit. This shows that
the new orbit is a simple periodic orbit of length p1 + p2, by Lemma
2.3. Now we fill in the x orbit (the longer orbit) and consider a torus
punctured at the y orbit and the new orbit together. We have the
family of vertical loops f i(A) and the action on it is exactly as in the
condition for a simple pair as the loop C can be isotoped to A plus a
loop around y0. We choose a horizontal loop as the loop D on Figure
19. Then its image D′ is as shown on Figure 19. The image has the
required properties. The orbit y together with the new orbit form a
simple pair for the homeomorphism f .
Next we fill in the y orbit and leave punctures at the x orbit and
the new orbit. We choose the initial vertical loop A differently, as
Figure 19.
in Figure 20. This loop A is one rectangle to the right of m plus a
loop on the left. After k = p2 − 1 iterations of f it will move to
line l which is q1 rectangles to the left of m. Indeed it will move to
q1(p2 − 1) + 1 = p1q2 − q1 rectangles to the right of m which means q1
rectangles to the left. The loop fk(A) looks like the loop A and lies
to the left of the point r3p2 and to the left of the point o2p1+4p2 . Next
iteration of f takes it to a curve which looks like f(l) but lies to the
left of op1+4p2+1 and to the left of o2p2+1. Since we filled the point y0 we
can isotop this loop to a vertical loop near m, which passes to the left
of op1+4P2+1. Subsequent iterations translate it to the right and f
p1(A)
is equal to curve B on Figure 20
Next p2− 1 iterations will take B to the loop near l which lies to the
left of o2p1+4p2 . Next iteration of f takes this loop to a loop similar to
f(l), but lies to the left of o2p2+1. Since the points of y-orbit are filled
we can isotop it to the loop C on Figure 20. It can be further isotoped,
relative to the x-orbit and the new orbit, to the loop A plus a small
loop around x0 to the left of A. If we choose the same horizontal loop
as in the previous case , with the same image as before, we get the
required action of f for a simple pair consisting of the x-orbit and the
new orbit.
Now we can continue by the same analysis for each of these two
simple pairs, finding their Farey intermediate to be a simple orbit as
well that forms a simple pair with each of them, and so on. It remains
to prove the persistence of all these simple pairs under isotopies. For
this, recall The following theorem from [11].
Figure 20.
Theorem 4.1. (Hall) Let S be a closed surface and let A be a finite
subset of S. Let f be a homeomorphism of S which leaves A invari-
ant. Let p = x1, . . . , xk be a finite collection of periodic points for f
which are essential, uncollapsible, mutually non-equivalent and non-
equivalent to points of A. Then the collection p is unremovable, which
means that for every homeomorphism g isotopic to f rel A there exists
an isotopy ft rel A and paths xi(t) in S such that f0 = f , f1 = g,
xi(0) = xi, xi(t) is a periodic point of ft of period equal exactly to the
period of xi.
(This theorem is a generalization of the main result of Asimov and
Franks in [2] to several periodic orbits. In fact this generalization was
mentioned in [2] as a remark with a hint of a proof.)
Recall also that if f is pseudo-Anosov in the complement of A then
it is condensed and by [6] Lemma 1 and Theorem 2.4 each periodic
point is uncollapsible and essential and points from different orbits are
non-equivalent and points disjoint from A are not equivalent to points
of A.
Corollary 4.2. : Let T be a torus and let A be a finite subset of T .
Let f be a shear-type homeomorphism of T which is pseudo-Anosov in
the complement of A. Let g be a homeomorphism of T isotopic to f in
the complement of A. If x is a simple periodic orbit for f then there
exists a simple periodic orbit z for g with ρ(z) = ρ(x). If x, y is a
simple pair of periodic orbits for f , one or both disjoint from A, then
there exists a simple pair of periodic orbits z, w for g with ρ(z) = ρ(x)
and ρ(w) = ρ(y).
Proof. Chose points x1 and y1 from the orbits x and y . By Theorem
4.1 there exists an isotopy ft and paths x1(t) and y1(t) such that x1(0) =
x1, y1(0) = y1, x1(t) is a periodic point of ft of a fixed order p for all
t and y1(t) is a periodic point of ft of a fixed order q for all t and
y1(t) = y1(0) for all t if y(0) ∈ A. For a given t all points in the orbits
of x1(t) and y1(t) for f are distinct, they form a braid with p+q strands.
They move when t changes and their movement can be extended to an
ambient isotopy ht which is fixed on A. Then ht(f
i(x1(0)) = f
t (x1(0))
and ht(f
i(y1(0)) = f
t (y1(0)). Consider isotopy Ft = h
t ftht. We have
i(x1(0)) = f
i+1(x1(0)) and Ft(f
i(y1(0)) = f
i+1(y1(0)) so Ft is fixed
on the orbits x and y. In particular x and y form a simple pair of
periodic orbits for F1 (or x forms a simple periodic orbit for F1 if there
is no y). But F1 = h
1 gh1 so h1(x) and h1(y) form a simple pair of
periodic orbits for g.
This concludes the proof of Theorem 1.1.
5. Global analysis of the kicked accelerated particle
system
The physical system called the kicked accelerated particle consists
of particles that do not interact with one another. They are subject
to gravitation and so fall downwards, and are kicked by an electro-
magnetic field, i.e., the electro magnetic field is turned on for a very
short time once in a fixed time interval. This electromagnetic field is a
sine function of the height of the particle, hence the particles are kicked
upwards or downwards by different amounts, depending on their po-
sition at the time of a kick. For a short review of the results for this
system see [8]. Experiments of this system were conducted by the Ox-
ford group, see [17], and the system was found to show a phenomena
that is now called ”quantum accelerator modes”: as opposed to the
natural expectation that particles fall with more or less the gravita-
tional acceleration, it was found that a finite fraction of the particles
fall with constant nonzero acceleration relative to gravity, as can be
seen in Figure 21
This is a truly quantum phenomenon having no counterpart in the
classical dynamics. A theoretical explanation for this phenomenon was
given by Fishman, Guanieri and Rebuzzini in [13], and it establishes a
correspondence between accelerator modes of the physical system, and
periodic orbits of the classical map
Figure 21. Accelerator modes
Experimental Data (taken from Oberthaler, Godun, d’Arcy, Summy
and Burnett, see [17]) showing the number of atoms with specified
momentum relative to the free falling frame as the system develops in
time (the numbers on the y axis represents time by the number of
kicks, while the z coordinate is proportional to the number of atoms)
J + k̃sin(θ + J) + Ω
θ + J
mod2π(1)
Where the J coordinate corresponds to the particles momentum, and
θ to its coordinate. This map is of shear type, and the acceleration for
a periodic orbit with rotation number q
is given by
− Ω(2)
Hence, by analyzing the structure of existence of periodic orbits for
the classical map above, we would be able to find which modes should
be expected for which values of the parameters k and Ω. We remark
that actual experimental observation also requires stability of the pe-
riodic orbits. It is important to stress here that since these parameters
correspond to the kick strength and the time interval between kicks
they can be controlled in the experiments as we wish, so results ob-
tained for this system can be tested experimentally. When one plots
the numerical results describing which periods exist for different values
of k and Ω one gets an extremely complicated figure, see figure 22.
Figure 22. Tongues of periodic orbits
An exact mathematical analysis of this system is extremely compli-
cated. Perturbative methods have been used in [13] to analyze the ex-
istence of these ”tongues” of periodic orbits in the region where k → 0,
as well as giving estimates on their widths.
Look at the map f given by (1) in regions where Ω is equal q
for some rational number q
in the unit interval, and small k. For a
small enough k it can be seen both from the numerical results shown
graphically in figure 22 and from perturbative arguments that in the
above region a periodic orbit with period p exists.
For small k the periodic points of this orbit must be pretty much
equally spaced along the J axis, and we can choose (for k small enough)
a family of vertical loops that are equally spaced at distance exactly Ω
apart, and each is at distance at least, say, 3k from any of the periodic
points.
The image of a loop parameterized by Γ1(θ) =
is given by(
J0 + ksin(J0 + θ) + Ω
J0 + θ
J0 + ksin(θ
′) + Ω
and so is very
close (for small k) to another loop of the chosen family. It follows that
there exists a map f̃ isotopic to f rel the orbit which keeps this family
of curves invariant, and so, by Lemma 2.3, all the periodic orbits seen
in the tips of the tongues in Figure 22 are simple orbits.
Note the rotation number of each of these orbits is equal exactly to
the value of Ω in the tip of the tongue (k = 0) as for very small k
the J coordinate increases by an almost fixed value, close as we wish
to Ω. And, by equation (2) the rotation number q
is related to the
acceleration of the corresponding acceleration mode by
α = q
2π − Ω
So the topological meaningful numbers here are in fact also the ones
with physical significance. While Ω changes through the region in which
this periodic orbit exists, q
is of course a topological invariant and
therefore fixed. Hence the acceleration vanishes on the line with fixed
Ω in the middle of each tongue, and changes signs when one crosses
this line. This was measured experimentally in [15].
For any other point higher in the tongue which we can reach by an
isotopy along which the periodic orbit exists, we also have the orbit
is a simple orbit. We will assume, as is very natural and was checked
numerically for many cases, that the orbits remain simple throughout
the region of each tongue.
In some of the cases for which we drew a portrait of the phase space,
we found that the fact the homeomorphism is isotopic to one which is
reducible rel the periodic orbit is realized by the physical map itself, as
seen in Figure 23.
Here the phase space is truly divided into pieces. Each of the annuli
in this decomposition is mapped to another, and returns to itself with
one twist after p iterations of f . Therefore every periodic orbit must
have a period which is a multiple of p. On the other hand, when an
annulus is mapped to itself with one twist under an area preserving
map (here under fp), every rotation number in the unit interval exists
for it (here we mean the standard annulus rotation number measuring
the rotations around the annulus), and so every period exists, as for
every rational number n
there is a periodic point of order m which
rotates n times around the annulus before it returns to itself. This
yields that for such a point in the parameter space, exactly all periods
that are multiples of p exist. It is our belief that this situation is typical
Figure 23. Phase portrait for a two-orbit
Drawn for k = 1
2π and Ω = π, the two-orbit which is clearly seen is
a stable orbit with two stable neighborhoods drawn. There is another
two-orbit present, at which the arrows point, and it is the stable and
unstable manifolds for this unstable orbit which divide the phase
space into non intersecting regions which do not mix.
for the center of each tongue, that is for Ω = 2π q
. At other points,
namely in all point we have numerically checked outside the center of
the tongue, orbits of coprime lengths may exist simultaneously. We
believe that the coexisting orbits whose rotation numbers are Farey
neighbors form a simple pair together, as in the example on Figure 24,
which shows a simple pair of orbits with rotation numbers 1/3 and 1/2
found in the physical system.
This coexistence happens at a point in Figure 22 for which two
tongues intersect. We assume that the same orbit persists through-
out the tongue, and therefore we have at such a point two coexisting
simple orbits. We believe that in all points of intersecting tongues
coming from k = 0 and Ω1 =
, Ω2 =
which are Farey neighbors,
p1 > p2, the coexisting orbits form a simple pair.
Theorem 3.7 therefore implies that there are infinitely many periodic
orbits for the parameters at a region of intersection of two such tongues,
with rotation numbers equal to all rational numbers between the ones
Figure 24. A pair of coexisting orbits in the physical system
Drawn with a collection of curves on the torus and their images,
which show this is a simple pair.
of these two tongues. If we assume all these simple orbits present also
come from tongues, this yields that each rational tongue between q1
and q2
intersects each of these two tongues lower (along the k axis) than
they intersect each other. In other words, following a path from a tip of
a tongue upwards in the tongue, if it intersects a Farey neighbor tongue
we know it intersects earlier all tongues of rational numbers between
them. This determines the global structure appearing in Figure 22 of
all accelerator modes in the physical system, as Sharkovskii’s theorem
determines it for one dimensional systems.
References
[1] R. L. Adler, Symbolic dynamics and Markov partitions, Bulletin of the AMS
35 N1 (1998), 1-56.
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in dynamics, Contemp. Math. 246 (1999), 17-45.
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Thurston, Cambridge University Press, 1988.
[8] M.B. d’Arcy, G.S. Summy, S. Fishman and I. Guarneri, Novel Quantum
Chaotic Dynamics in Cold Atoms, Physica Scripta 69 (2004), 25-31.
[9] E. Doeff, Rotation measures for homeomorphisms of the torus homotopic to a
Dehn twist, Ergod. Theor. Dynam. Syst. 17 (1997), 1-17.
[10] E. Doeff and M. Misiurewicz, Shear rotation numbers, Nonlinearity 10 (1997),
1755-1762.
[11] T. Hall, Unremovable periodic orbits of homeomorphisms, Math. Proc. Camb.
Phil. Soc. 110 (1991), 523-531.
[12] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers,
Clarendon Press, Oxford, 1979.
[13] S. Fishman, I. Guanieri and L. Rebuzzini, J. Stat. Phys. 110 (2003), 911; S.
Fishman, I. Guarneri and L. Rebuzzini, Phys. Rev. Lett. 89 (2002), 84101-1-
4; I. Guarneri, L. Rebuzzini and S. Fishman, Arnol’d Tongues and Quan-
tum Accelerator Modes, submitted for publiaction in Nonlinearity, (quant-
ph/0512086)
[14] M. Misiurewicz, Rotation Theory in: Online Proceedings of the RIMS Work-
shop on ”Dynamical Systems and Applications: Recent Progress”.
[15] Z-Y. Ma, M.B. d’Arcy and S. Gardiner, Phys. Rev. Lett. 93 (2004), 164101-1-4.
[16] T. Matsuoka, Braids of periodic points and a 2-dimensional analogue of
Sharkovskiis ordering, World Sci. Adv. Ser. in Dynamical Systems 1 (1986),
5872.
[17] M.K. Oberthaler, R.M. Godun, M.B. d’Arcy, G.S. Summy, and K. Burnett,
Phys. Rev. Lett. 83 (1999), 4447-4451.
Tali Pinsky
Department of Mathematics
The Technion
32000 Haifa, Israel
e-mail: otali@tx.technion.ac.il
Bronislaw Wajnryb
Department of Mathematics
Rzeszow University of Technology
ul. W. Pola 2, 35-959 Rzeszow, Poland
e-mail: dwajnryb@prz.edu.pl
http://arxiv.org/abs/quant-ph/0512086
http://arxiv.org/abs/quant-ph/0512086
	Abstract
	1. Introduction
	2. Simple orbits 
	3. Simple orbit pairs
	4. The order relation
	5. Global analysis of the kicked accelerated particle system 
	References

  Sharkovskii proved that the existence of a periodic orbit in a
one-dimensional dynamical system implies existence of infinitely many periodic
orbits. We obtain an analog of Sharkovskii's theorem for periodic orbits of
shear homeomorphisms of the torus. This is done by obtaining a dynamical order
relation on the set of simple orbits and simple pairs. We then use this order
relation for a global analysis for a quantum chaotic physical system called the
kicked accelerated particle.

Introduction
Given a dynamical system (X, f), a key question is which periodic
orbits exist for this system. Since periodic orbits are in general difficult
to compute, we would like to have the means to deduce their existence
without having to actually compute them.
Sharkovskii addressed the dynamics of continuous maps on the real
line. He defined an order C on the natural numbers, Sharkovskii’s or-
der (see [14]), and proved that the existence of a periodic orbit of a
certain period p implies the existence of an orbit of any period q C p.
We say the q orbit is forced by the p orbit. This offers the means of
showing the existence of many orbits if one can find a single orbit of
“large” period. For a dynamical system depending on a single parame-
ter, if periodic orbits appear when we change the parameter, they must
appear according to the Sharkovskii’s order. Hence, Sharkovskii’s the-
orem gives the global structure of the appearance of periodic orbits for
one dimensional systems. Ever since the eighties there has been inter-
est in obtaining analogs for Sharkovskii’s theorem for two dimensional
systems (see [4] and [16]).
A homeomorphism of a torus is said here to be of shear type if it
is isotopic to one Dehn twist along a single closed curve. Let h be a
shear homeomorphism, and let x be a periodic orbit of h. We can then
define the rotation number of x, see discussion in Section 2. Thus, a
rational number in the unit interval [0, 1) is associated to each orbit.
We consider orbits up to conjugation: orbits (x, f) and (y, g) are
similar (of the same type) if there exists a homeomorphism h of the
torus T 2 such that h is isotopic to the identity, h takes orbit x onto
orbit y and hfh−1 is isotopic to g rel y. We define below a specific
family of periodic orbits we call simple orbits. In this family there is a
unique element up to similarity corresponding to each rotation number;
hence they can be specified by their rotation numbers. We emphasize
it is not true in general that an orbit of a shear homeomorphism is
characterized by its rotation number.
Simple orbits are analyzed in Section 2. As it turns out (see Remark
in section 2), one simple orbit is indeed simple and does not force the
existence of any other orbit. More generally a periodic orbit is of twist
type if it does not force the existence of any orbit of different type with
the same rotation number. It is tempting to conjecture that the simple
orbits are the only orbits of twist type, but Lemma 2.4 shows that this
is false. We give there an example of an orbit of twist type which is
not simple. This example also shows that a periodic orbit with a given
rotation number does not necessarily force a simple orbit of the same
rotation number.
We turn in Section 3 to analyze pairs of orbits. Two coexisting simple
periodic orbits can form a simple pair and these are considered. The
pairs do force some more interesting dynamics, as follows. We denote
the integers by letters p, q and the rational numbers by r, s, t possibly
with indices. For a pair of simple orbits of rotation numbers q1
and q2
to constitute a simple pair, it is necessary that the rotation numbers
be Farey neighbors, i.e. |p2q1 − p1q2| = 1. We denote such a pair of
rational numbers by q1
We now define an order relation on the following set P of rational
numbers and pairs in the unit interval,
P = {r|r ∈ Q ∩ [0, 1)} ∪ {r ∨ s|r, s ∈ Q ∩ [0, 1)}.
Define the order relation on P to be
r ∨ s < t⇔ t ∈ [r, s]
r1 ∨ r2 < s1 ∨ s2 ⇔ s1, s2 ∈ [r1, r2].
where we denote by [r1, r2] the interval between r1 and r2, regardless
of their order.
Theorem 1.1. The order relation < on P describes the dynamical
forcing of simple periodic orbits. Namely, the existence of a simple pair
of periodic orbits with rotation numbers r∨s in P implies the existence
of all simple orbits and simple orbit pairs of rotation numbers smaller
than r ∨ s according to this order relation .
This is the main result of this paper and the proof is completed in
Section 4. The idea of the proof is as follows. Consider the torus punc-
tured on one or more periodic orbits of a homeomorphism h. Then h
induces an action on this punctured torus, and on its (free) fundamen-
tal group. Now apply the Bestvina-Handel algorithm to this dynamical
system. The idea of using the Bestvina-Handel algorithm was used by
Boyland in [6] and he describes the general approach in [5]. In our
case, after puncturing out a simple pair of orbits, applying the algo-
rithm yields an isotopic homeomorphism which is pseudo-Anosov. The
algorithm also offers a Markov partition for this system and we use
the resulting symbolic representation to find that there are periodic
orbits of each rotation number between the pair of numbers we started
with. Then we directly analyze the structure of the pseudo-Anosov
representative to show that all these orbits are in fact simple orbits.
Furthermore any two of them corresponding to rotation numbers which
are Farey neighbors form a simple pair. Finally we establish the iso-
topy stability of these orbits using results of Asimov and Franks [2] and
Hall [11]. Thus, the orbits exist for any homeomorphism for which the
simple pair exists, and are forced by it.
One should compare this result with a very strong theorem of Doeff
(see Theorem 3.6), where existence of two periodic orbits of different
periods for a given shear homeomorphism h implies existence of peri-
odic orbit of every intermediate rotation number. However an explicit
description of these orbits is not given, while our stronger assumptions
imply existence of simple periodic orbits and simple pairs of orbits. It
may be true that the existence of any two periodic orbits with differ-
ent rotation numbers implies the existence of a simple orbit with any
given intermediate rotation number, but we feel that the evidence is not
strong enough to make a conjecture either way, in particular in view
of Lemma 2.4. Even more difficult question is to determine whether
there exists a simple pair of periodic orbits in the situation of Doeff
Theorem. First one should try to find a pair of simple orbits which is
not a simple pair while the rotation numbers are Farey neighbors, but
it is very difficult to understand the geometry of the pseudo-Anosov
homeomorphism which arises in this situation.
This research was originally motivated by a question we were asked
by Professor Shmuel Fishman: Is there a topological explanation for
the structure of appearance of accelerator modes in the kicked particle
system. In section 5 we give a description of the kicked particle system.
This system turns out to be described precisely by a family of shear
homeomorphisms of the torus. The existence of accelerator modes is
equivalent to existence of periodic orbits. The global structure of this
system is given by the order relation in Theorem 1.1, while it cannot
be directly computed due to the complexity of the system.
The authors would like to thank Professor Shmuel Fishman for offer-
ing valuable insights, Professor Italo Guannieri for some critical advice,
and Professor Philip Boyland for many indispensable conversations.
2. Simple orbits
Let {x1, ....., xN} be a set of points belonging to one or more pe-
riodic orbits for a homeomorphism f of a surface S. The dynamical
properties of this set of orbits are captured by the induced action of f
on the complement S0 = S \ {x1, ...., xN} in a sense that will shortly
become clear. Choose any graph G which is a deformation retract of
the punctured surface S0. A homeomorphism of S0 then induces a map
on G. The converse is also true: a given map of G determines a homeo-
morphism of S0 up to isotopy. Therefore we specify the periodic orbits
we analyze in terms of the action on a graph which is a deformation
retract of the surface after puncturing out the orbit.
Denote by GN the graph obtained by attaching N small loops to the
standard unit circle at the points exp(2πj
i), j = 0, . . . , N − 1.
Definition 2.1. We call a periodic orbit x = {x1, . . . , xN} for a shear
homeomorphism f on the two-torus a simple orbit if the following hold.
(1) There can be found a graph G which is a deformation retract
of T0 = T
2 \ x as on Figure 1 such that G is homeomorphic to
We call the loop in G corresponding to the unit circle the hor-
izontal loop and the loops attached to it the vertical loops.
(2) There exists a homeomorphism f̃ of T0 isotopic to f rel x (i.e.
the isotopy is fixed on x) so that a neighborhood of G is invari-
ant under f̃ and the induced action on G satisfies: (a) There
exists a fixed number k ∈ {0, .., N − 1} such that each vertical
loop is mapped k loops forwards (clockwise along the unit cir-
cle) to another vertical loop. (b) The horizontal loop is mapped
to itself with one twist around one of the vertical loops.
Figure 1. A standard graph for a simple orbit
Figure 2. The action on a standard graph for a simple orbit
Remark. A homeomorphism h for which we are given a simple periodic
orbit must be of shear type as we can deduce from the action on the
homology of the non-punctured torus.
For a shear homeomorphism there exists a basis for the first homol-
ogy for which the induced map is represented by the matrix
From here on we refer to any two axes given by an homology basis that
gives us the above representation as standard axes. The horizontal loop
and one of the vertical loops in a graph for a simple orbits constitute
a standard basis.
Definition 2.2. Let h be a shear homeomorphism, and let ĥ be a lift
of h to the universal cover (a plane). For any periodic point x of h
of period p, ĥp maps any lift x̂ of x the same integer number q along
the horizontal axis away from x̂, in a standard choice of axis (and x̂ is
possibly mapped some integer number along the vertical axis as well).
We can then define the rotation number of x to be ρ(x) = q
mod 1.
The rotation number does not depend on the lift ĥ of h.
(The rotation number is often define relative to the given lifting of
the homeomorphism h and is not computed modulo 1, but we want it
to depend only on the orbit and not on the lifting. In particular we
want a simple orbit to have well defined rotation number independent
of the lifting of h.)
Remark. In the case of a homeomorphism isotopic to a Dehn twist on a
torus, which is our interest here, it can be easily shown that the abelian
Nielsen type equals exactly the rotation number defined above.
There exists a simple orbit for any given rational rotation number
r ∈ [0, 1), and it is unique up to similarity. Denote the similarity class
by r̂. In the following we use the word vertical to describe the y axis,
in a standard choice of axis for f (the direction along which the twist
is made).
Lemma 2.3. Let x be a periodic orbit for a shear-type homeomorphism
f of T2 for which there exists a family of vertical loops such that they
bound a set of annuli each containing one point of the periodic orbit,
and this family is invariant under a homeomorphism f̃ isotopic to f
rel x. Then x is a simple orbit.
Proof. Choose a vertical loop l of the invariant family. f is orientation
preserving, and so is f̃ . The first loop to the right of l is therefore
mapped to the first loop to the right of f̃(l). Hence, the vertical loops
in the invariant family are all mapped the same number of loops to the
right.
Now we have to find a horizontal line with the desired image. We
write the invariant family of loops as {li}
i=1, where p = period(x),
ordered along the horizontal axis. We choose another family of vertical
loops {mi}
i=1, such that mi is contained in the annulus between li and
li+1 (lp+1 = l1), and passes through the periodic point xi also contained
in this annulus. Choose a point a1 6= x1 on m1. f can be adjusted in
such a way that the new homeomorphism f̃ leaves both families of
vertical loops invariant, and in addition, so that a1 be a periodic point
of f̃ with period p. We denote the orbit of a1 by {ai}
i=1 where ai ∈ mi
for 1 ≤ i ≤ p.
Choose a line segment n1 connecting a1 to a2, so it crosses the annulus
between m1 and m2 from side to side. We choose nj+1 to be the line
segment f̃ j(n1) for 1 ≤ j ≤ p − 2. The boundary points of nj and nk
coincide whenever they lie on the same vertical loop.
Now, we look at the horizontal loop n =
j=1 nj. Each segment of n
is mapped exactly to the next segment, except np−1 which is mapped
into the annulus between m1 and m2. Since the mapping class group
of an annulus is generated by a twist with respect to any loop going
once around the annulus we may assume, that np is mapped to n1 plus
a number of twists along such a loop. On the other hand we know that
f(n) is homotopic to itself plus one twist in the negative direction (on
the closed torus), so f̃ maps n to itself plus one negative twist along
this loop. By further adjustment of f̃ we may assume the twist is made
along l2. Thus the union of the vertical family {li} with n chosen as
above constitute a graph showing x to be a simple orbit. �
The Thurston-Nielsen classification theorem, see [7], states that any
homeomorphism f on a closed connected oriented surface of negative
Euler characteristic is isotopic to a homeomorphism f̃ which is
(1) pseudo Anosov, or
(2) of finite order, or
(3) reducible.
where a homeomorphism φ is called of finite order if there exists a
natural number n such that φn = id. A homeomorphism φ is called
pseudo-Anosov if there exists a real number λ > 0 and a pair of
transverse measured foliations (Fu, µu) and (F s, µs) with φ(Fu, µu) =
(Fu, λµu) and φ(F s, µs) = (F s, 1
µs). A homeomorphism φ on a sur-
face M is called reducible if there exists a collection of pairwise disjoint
simple closed curves Γ = {Γ1, ...,Γk} in int(M) such that φ(Γ) = Γ
and each component of M \Γ has a negative Euler characteristic. The
representative f̃ in the isotopy class of f which is of one of the three
forms above is called the Thurston-Nielsen canonical form of f .
When the surface has a finite number of punctures and φ permutes
the punctures then the same is true except that in the case of pseudo-
Anosov map we treat the punctures as distinguished points (there is a
unique way to extend a homeomorphism to the distinguished points)
and we allow an additional type of singularities of the measured foli-
ations, the 1-prong singularities at the distingushed points (See [11],
and section 0.2 of [3]).
Of course homeomorphism f is reducible with respect to a simple
orbit since it contains an invariant family of loops {li} and the comple-
ment of the invariant family consists of punctured annuli (which have
negative Euler characteristic).
Remark. A homeomorphism f with a simple orbit x can be constructed
in such a way that x is the only periodic orbit of f . The invariant
set of vertical loops is evenly spaced with the distance between the
consecutive loops equal to 1/p. The loops are moved by q/p to the
right and by fixed irrational number downward. Punctures (the points
of the periodic orbit) are also evenly spaced and have the same height.
The vertical lines containing punctures are moved by q/p to the right.
The punctures keep their height and all other points of the loop move a
little downwards. Every other vertical line is moved to another vertical
line by a little more than q/p to the right (not all lines by the same
distance).
Example 1. Not every periodic orbit for a shear homeomorphism is
reducible. Consider the homeomorphism h described on Figure 3. It
takes the graph on the left of Figure 3 to the graph on the right and is
a shear homeomorphism. It has a periodic orbit of order 2, shown on
the pictures, with rotation number 1/2 and it is pseudo-Anosov in the
complement of the orbit.
Figure 3.
Lemma 2.4. There exists an orbit of twist type for a shear homeomor-
phism of the torus which is not simple.
Proof. We construct an example of an orbit of length 4 with the rota-
tion number 1/2. It cannot be a simple orbit and yet we prove it does
not force the existence of any periodic orbit not similar to itself, and is
thus of twist type. Such examples may be known, possibly considered
for a different phenomena. We include it here in order to show the
independence of our results.
We represent the torus as the unit square with the opposite sides
identified. The points A1, A2, A3, A4 of the orbit are spaced evenly on
the horizontal middle line with the x-coordinate 1/8, 3/8, 5/8, 7/8.
We split the square into 2 equal parts U1 and U2 by the vertical line
x = 1/2. Homeomorphism h translates U1 to the right to U2. Vertical
lines go to vertical lines, lines x = 0 and x = 1/2 move downward by
an irrational number α < 1/40 and the movement is damped out to the
horizontal translation for t < α and t > 1/2− α, so the other vertical
1 2 3 4 5 6 7
Figure 4. Some vertical lines in U2
� �' $
'' %%%�
' %' %��
%' %' %�
1 2 3 4 5 6 7
Figure 5. The foliation in U2 realizing a non-simple
orbit of twist type.
lines are translated horizontally by 1/2. In particular A1 moves to A3
and A2 moves to A4.
The restriction of h to U2 is defined in two steps. The second step
simply translates U2 horizontally by 1/2 to the right (which is the
same as the translation by 1/2 to the left). The first step is isotopic
to the half-twist along the segment connecting A3 and A4, followed by
the Dehn twist with respect to the right side (right boundary of the
cylinder). In particular it switches A3 and A4. We shall prove that we
can construct such h for which h2 has no fixed points and therefore h
has no periodic orbit of length 2 and in particular no simple orbit with
the rotation number 1/2.
We describe the first step of h restricted to U2. Figure 4 shows some
vertical lines in U2, the big dots show the points A3 and A4 of the
periodic orbit. Figure 5 shows their images under the first step. In
these pictures U2 is represented as a square, to make more space, but
in the reality the base has length 1/2 and the height is equal to 1. Line
x = 1/2 is mapped to itself and moves downward by α. The near by
vertical lines (for t < 1/2+1/30) are moved to the vertical lines and to
the right where line x = 1/2+1/30 is moved to the line x = 1/2+1/20.
For t ∈ (1/2 + 1/30, 1−1/20) the line x = t is moved to a curve Lt and
for t ∈ (1−1/20, 1) the line x = t moves to a vertical line to the right of
it and downward, to get the full Dehn twist plus a movement downward
by α when we get to the line x = 1. For t ∈ (1/2+1/30, 1−1/20) curve
Lt starts at a point on the top side to the right of x = t, it moves to the
left, then to the right, then to the left again and ends at the bottom
side (exactly below its starting point). In particular each vertical line
meets Lt in at most two points. Some lines Lt are shown on Figure 5.
We may arrange it in such a way that there exist t0, t1 such that
1/2 < t0 < t1 < 1 and the line x = t:
is disjoint from Lt, and lies on the left side of Lt when t < t0;
meets Lt at one point for t = t0;
meets Lt at two points when t0 < t < t1;
meets Lt at one point for t = t1;
is disjoint from Lt and lies on the left side of Lt when t1 < t < 1.
We get a new trivial foliation of the annulus U2. In step 1 we map
the vertical foliation onto the new foliation Lt. We can further change
the first step moving each leave Lt along itself to reach the following
goal. Let Pt, Qt denote the intersection points of x = t with Lt, Pt
lies below Qt (the points coincide for t0 and t1). For t = t0 the line
x = t meets Lt in one point Pt. We may assume that the image of Pt
in Lt lies in the part below Pt. Then for the nearby leave the images
of both points Pt and Qt lie in the lower part of Lt below the point
Pt (see the small dots on the first curve in Figure 5). The images of
Pt and Qt lie further away from each other when we move to the right
(see the small dots on the second curve). The third line passes through
A3, its image Lt passes through A4 and the images of Pt and Qt lie on
different sides of A4 along the third curve. Next the upper point Qt
moves backwards along Lt and when we reach the fourth line of Figure
4 (also shown on Figure 5) it coincides with the point Pt on Lt. Next
the image of Qt lies inside the arc of Lt between the points Pt and Qt
and when we reach the fifth line on Figure 4, which passes through the
point A4, then the curve Lt passes through A3 and the image of Qt on
Lt lies above A3 (see Figure 5). When we move further to the right
the image of the point Qt moves again forward towards the image of Pt
and at the line number 6 on Figure 4 the image of Qt again coincides
with Pt at the intersection of x = t with Lt. Next the images of Pt and
Qt move further down and gets close together and when t = t1 we have
one intersection point Pt and its image lie below Pt along Lt. Step 1
has no fixed points. Step 2 translates U2 to U1.
We now consider the homeomorphism h2. We start with U1. Any
point on x = 0 and x = 1/2 moves down by 2α. Any point with
x ∈ [0, 1/30] moves to a point with a bigger x-coordinate. Any point
with x ∈ [1/30, 9/20] moves horizontaly by 1/2 then we apply step 1,
which has no fixed points, and then the point moves again horizontaly
by 1/2 so it comes to a new point. Any point with x ∈ [9/20, 1/2]
moves to a point with a bigger x-coordinate.
For points in U2 the situation is similar. Any point with x ∈ (1/2, 16/30]
moves to a point with a bigger x-coordinate. Any point with x ∈
[16/30, 19/20] moves under the first step to a new point with the x-
coordinate in [11/20, 29/30] and then moves horizontaly twice by 1/2.
Finally any point with x ∈ [19/20, 1] moves to a point with a bigger
x-coordinate. Homeomorphism h2 has no fixed points and h has no
periodic points of order 2.
We now show that there exists a homeomorphism f isotopic to h
in the complement of the orbit A1, A2, A3, A4, which has only periodic
orbits similar to this orbit and periodic orbits of order 2. We consider
parts U1 and U2 as before. The restriction of f to U1 translates it hor-
izontaly by 1/2. In U2 we choose two circles with center (3/4, 1/2) and
radius 1/7 and 1/6 respectively. We rotate the interior of the smaller
circle by 180 degrees. The rotation is damped out to the identity at
the outer circle and the intermediate circles are moving out towards
the outer circle. The exterior of the outer circle with x < 19/20 is
pointwise fixed. The lines with x > 19/20 move to the right and down
to get the full Dehn twist when we get to the line x = 1. The second
step of f restricted to U2 translates it horizontally by 1/2. Now each
point inside the smaller circle, different from its center (which has pe-
riod 2), belongs to an orbit similar to A1, A2, A3, A4. Points between
the circles and points with x ∈ (19/20, 1) are not periodic and other
points in U2 have period 2 and the same is true for the corresponding
points in U1.
Therefore the orbit A1, A2, A3, A4 does not force any periodic orbit
not similar to itself. �
3. Simple orbit pairs
Let x and y be two coexisting simple periodic orbits, for a homeo-
morphism f of T2 (f must be of shear type), with rotation numbers q1
and q2
respectively. Assume p1 > p2, i.e., y has lesser period than x.
Definition 3.1. We call the pair of orbits a s imple pair if
• We can find an embedded graph G in their complement home-
omorphic to Gp1 as on Figure 6.
Figure 6.
Each component in the complement of the graph is a topolog-
ical rectangle which contains exactly one point of orbit x and
at most one point of orbit y.
• The homeomorphism f acts on this graph in the following way:
each vertical loop except one moves to another vertical loop,
there is one vertical loop denoted l such that f(l) is a vertical
loop m plus a small loop around one point of the (shorter) y
orbit, in a rectangle adjacent to line m on the right (as on Figure
7) or on the left, and the horizontal line is mapped to itself plus
a twist in the negative direction around f(l), as on Figure 7.
The graph which appears in Definition 3.1 divides the torus T2 into p1
rectangles. The homeomorphism f moves each vertical loop the same
Figure 7.
distance, say k rectangles, to the right except for the small additional
loop for line l. Let R0 be the rectangle adjacent to m in which the
small loop in the image of the graph occurs. R0 must contain exactly
one point of each orbit. We denote these points x0 and y0 respectively.
Under p1 iterations of f the point x0 runs q1 times around the whole
torus, that is q1p1 rectangles to the right. So, p1k = q1p1 and k = q1.
The point y0 is mapped to itself after p2 iterations.Under each itera-
tion the image of y0 is mapped k (= q1) rectangles to the right, except
the last iteration under which it is moved an additional rectangle to the
right or left. Altogether it has moved p2q1± 1 rectangles. At the same
time, it is mapped around the torus q2 times, hence q2p1 rectangles.
This means p2q1 ± 1 = q2p1. Thus for a simple pair of periodic orbits
the rotation numbers q1
and q2
are Farey neighbors and the additional
loop is on the right (as on Figure 7) if and only if q2
. Denote
by r̂ ∨ ŝ the similarity class of a simple pair corresponding to a pair of
Farey neighbors r ∨ s.
Consider again the points x0 and y0 in the rectangle R0. Continue
the notation to all the points of x and y by xi = f
i(x0) and yi = f
i(y0).
We draw a small loop around each of the points of y. The union of
these loops will be the peripheral subgraph P for the Bestvina-Handel
algorithm, since we may assume the union of these loops to be f -
invariant. Now we consider separately two cases. Case 1 will be the
case in which m is the left boundary curve of R0, while in case 2 it is
the right boundary (in other words in the first case q1
and in the
second case q1
). Choose some point on the loop around y0 and
connect it, by a curve l0, to a point on the section of the horizontal line
in R0, in case 1 from below the segment and in case 2 from above.
Figure 8.
Then f(l0) is a curve connecting the loop around y1 and the corre-
sponding horizontal segment. We denote it by l1, and do the same for
each yi. After adding the above edges to the graph case 1 is topologi-
cally as in Figure 8.
The inclusion G ↪→ S0 is a homotopy equivalence (where S0 is the
punctured torus). We know the action of f on all edges of G except
for the curve lq1−1 connecting yq1−1 and the horizontal segment in the
corresponding rectangle. It’s image is a curve connecting the horizontal
segment in the rectangle adjacent to m which is not R0 to the loop
around y0. This image might wind around a disk containing y0 and
xq1−1 as in Figure 9
Figure 9.
The graph and its image for case 2 are exactly the same except that
the loops are connected to the horizontal segments from above. We
shall prove in Proposition 3.2 that we may assume that the image of
the segment lq1−1 has no winding. Hence we draw from now on the
graph images without winding, and we may assume the graphs given
in Figure 10 also have an invariant neighborhood by a further isotopy
of f .
Figure 10. The standard graph for a simple pair, case
1 on the left and case 2 on the right.
The action of f (up to isotopy) on this graph is given by one of the
actions on Figure 11, drawn in some regular neighborhood of the graph,
where each vertical loop moves q1 loops to the right.
Figure 11. The action on a standard graph for a simple
pair, for both cases respectively.
Proposition 3.2. Let {x, y} be a simple pair with the graph as on
Figure 8 and with windings as on Figure 9. Then there may be chosen a
different invariant graph, which also makes {x, y} a simple pair, whose
image is without winding.
Proof. To simplify the picture we prove the proposition for rotation
numbers 1
and 1
. The general proof proceeds in the same way. We
start by looking at a simple pair {x, y} of rotation numbers 1
and 1
a homeomorphism f with the corresponding invariant graph G given
so that the action on it is without any twists, as on the left side of Fig-
ures 10 and 11. We now choose a different system of curves (a different
graph), in a small neighborhood of G as in Figure 12, which will serve
as a new graph for the pair.
Figure 12.
Solid lines are the new vertical loops and dashed lines are the new di-
agonal segments like in figure 10. The horizontal loop consists of the
dashed lines and the long pieces of the solid lines. To move from left
to right along the horizontal line, move along a dashed line and turn
to the left when meeting a solid line. Continue up along a vertical loop
and then along the next dashed line. We add to this graph the periph-
eral subgraph and the connecting segments and get the graph H as on
Figure 13. It is clear that topologically the graph H has the same form
as the graph on Figure 10 and that it has an invariant neighborhood.
The reader can check (using the precise knowledge of the image of
each edge of the original graph) that the action of f on the graph H
has the properties required from a simple pair. Each vertical loop is
mapped onto another vertical loop except for one loop l for which f(l)
is equal to a loop m plus a loop around the next periodic point yp
of the shorter orbit. The horizontal loop is mapped onto itself plus a
negative Dehn twist along f(l). Consider the image s of the segment
which connects the horizontal loop to the periodic point yp−1. When
the action has no twist then s moves along the horizontal loop in its
Figure 13.
positive direction until it meets the original segment connecting to yp
and then it follows along the segment. However in our case s goes first
backwards along the horizontal loop than moves in the counterclockwise
direction along the boundary of the ”rectangle” adjacent to the vertical
loop m and finally follows the horizontal loop and the segment to yp.
This means that the action f on the graph H has one positive twist.
We proved that a simple pair for a shear homeomorphism with a
given graph and a given action without twists can be given another
graph which also describes it as a simple pair and the action on the new
graph has one positive twist. This process is reversible. Therefore, by
induction, we can add or remove any number of twists using a suitable
graph. This implies Proposition 3.2. �
Hence for a simple pair the action on a spine is given by Figures 10
and 11. We can now apply the Bestvina-Handel algorithm (see [3]),
endowing a neighborhood of G with a fibered structure in the natural
way. The algorithm specifies a finite number of steps which we apply
to the graph G, altering G together with the induced action on it, but
without changing the isotopy class of f on T2 \ (x ∪ y). When the
algorithm terminates, it gives a new homeomorphism f̃ which is the
Thurston Nielsen canonical form of f .
For simple pairs, the action in each of the two cases above is easily
seen to be tight, as no edge backtracks and for every vertex there are
two edges whose images emanate in different directions. The action
has no invariant non-trivial forest or nontrivial invariant subgraph and
the graphs have no valence 1 or 2 vertices. This is the definition in [3]
for an irreducible map on a graph.
Definition 3.3. Assuming g, the induced map on the graph itself, does
not collapse any edges, there is an induced map Dg, the derivative of
g, defined on
{(v, e)|v is a vertex of G, e is an oriented edge emanating from v}
by Dg(v, a) = (g(v), b) where b is the first edge in the edge path g(a)
which emanates from g(v).
Definition 3.4. We say two elements (v, a) and (v, b) in L corresponding
to the same vertex v are equivalent if they are mapped to the same
element under D(gn) for some natural n. The equivalence classes are
called gates
The gates in each of the cases above are given by Figure 14, indicated
there by small arcs. There is no edge which g sends to an edge path
Figure 14.
which passes through one of the gates - enters the junction through one
arm of the gate and exits through the other. Such an irreducible map
is efficient. i.e., this is an end point of the algorithm. Now, since there
are edges mapped to an edge path longer than one edge, we arrive at
our next theorem.
Theorem 3.5. A homeomorphism f of the two torus for which a simple
pair of periodic orbits exists is isotopic to a pseudo-Anosov homeomor-
phism relative to this pair of orbits.
Let f be a shear type homeomorphism of the torus, and fix a lift f̃
of f . Define the lift rotation number of a point x ∈ T2 to be
ρ(x, f̃) = limn→∞
(f̃n(x̂)− x̂)1
for any lift x̂ of x, when the limit exists, where the subscript 1 denotes
the projection to the horizontal axis. Define the rotation set ρ(f̃) of f̃
to be the set of accumulation points of{
(f̃n(x̂)− x̂)1
|x̂ ∈ R2and n ∈ N
Then, the above theorem follows from the following much more gen-
eral theorem by Doeff, see [9] and [10].
Theorem 3.6. (Doeff) Let h be a shear type homeomorphism of T2,
and fix a lift h̃ of h. If h has two periodic points x and y with ρ(x, h̃) 6=
ρ(y, h̃) then h is pseudo-Anosov relative to x and y. Furthermore, the
closure of the rotation set is a compact interval, and any rational point
r in the interior of this interval corresponds to a periodic point x ∈ T2
with ρ(x, h̃) = r.
In particular, Doeff proves existence of two periodic orbits of dif-
ferent rotation numbers implies existence of an orbit for any rational
rotation number between these two. But he does not give any charac-
terization of these orbits. Example 1 shows two different orbits, both
with rotation number equal to 1/2, one of which is pseudo-Anosov,
and the other reducible. Thus the rotation number does not give much
information about the orbit and in this sense this theorem does not
give a satisfactory dynamical understanding of what is happening in
regions of coexistence of orbits. In contrast with Doeff’s general theo-
rem, we get results for a very specific family of periodic orbits, but for
this family we are able to give exactly the orbits forced by others, as
we show in section 4.
In our case the canonical form f̃ of f we get by applying the Bestvina-
Handel algorithm is a pseudo Anosov homeomorphism. When this is
the case, the algorithm gives a canonical way of endowing a regular
neighborhood S0 of G with a rectangle decomposition {R1, ..., RN}.
The decomposition is a Markov partition for the homeomorphism PSL.
A Markov partition for a dynamical system offers a symbolic repre-
sentation for the system in the following way. Let ΣN be the subset of
the full N -shift (the set of bi-infinite series on N symbols), where N is
the number of rectangles in the decomposition. Let Σ be a subset of
ΣN defined by
Σ = {s = (..., sn, sn+1, ....) : Rsn ∩ f̃−1Rsn+1 6= ∅}
On Σ we naturally define a dynamical system with the operator of the
right shift denoted by σ, and (Σ, σ) is called the subshift corresponding
to the dynamical system. Σ can be completely described by stating
which transitions k → m for k,m ∈ {1, ...., N} are allowed (i.e., for
which k,m, f̃−1Rm ∩ Rk 6= ∅). See [1] for the definitions and for a
proof that in this case we can define a map π : Σ→ S0 by
π : s 7→
n=0 f̃
nRs−n ∩ .... ∩ f̃−nRsn
which satisfies the following properties:
• πσ = f̃π,
• π is continuous,
• π is onto.
We take here the set of sequences with the Tichonoff topology. Thus
a periodic point in the symbolic dynamical system which is just a peri-
odic sequence corresponds to a periodic point in the original dynamical
system.
To obtain the Markov partition in our case as in [3], we thicken the
edges of the graph to rectangles. In particular, the rectangles can be
glued directly to each other without any junctions. This can be done
in a smooth way, endowing S0 with a compact metric space structure
by giving a length and width to each rectangle, consistently. Each edge
of the standard graph for the pair (figure 10) corresponds to one rec-
tangle, except the edge which is mapped to the loop around y0. This
edge we divide in two (this is necessary to avoid having a rectangle
intersecting twice an inverse image of another rectangle). Now we have
edges of 7 different types on the graph. The vertical loops of the graph
consist of long edges we denote as A edges, and short edges we call
B edges. The loops around the points of the y orbit and vertical seg-
ments connecting the loops to the diagonal edges we call C’s and D’s
respectively. In rectangles which contain two punctures and therefore
two diagonal edges we call the upper ones L edges and the lower ones
K edges in the first case, and the lower ones L edges, upper ones K
edges, in the second case. The last type of edges are diagonals of once
punctured rectangles, these we call M edges.
Next, we label the rectangles in order to have explicitly the transition
rules:
• For 0 ≤ i ≤ p2 − 1 denote the rectangle corresponding to the
D edge connecting the loop around yi to the diagonal by ri+1.
Denote the rectangle corresponding to the C edge which is the
loop around yi by rp2+i+1.
• For 1 ≤ i ≤ p2, denote the rectangles corresponding to the L
and K edges connected to ri by r2p2+i and r3p2+i respectively.
Figure 15.
• Denote the rectangle corresponding to the A edge belonging to
the vertical line we referred to as m by r4p2+1 and the B edge
which is part of the same line m as r4p2+p1+1.
• For the vertical line f i(m) denote it’s A and B rectangles by
r4p2+1+i and r4p2+p1+1+i respectively for all 1 ≤ i ≤ p1 − 2
• For the vertical line fp1−1(m), denote it’s A edge as by r4p2+p1 .
There are two rectangles corresponding to the B edge as ex-
plained above, denote the lower one by r4p2+2p1 and the upper
one by r4p2+2p1+1.
• Label the p1− p2 remaining rectangles corresponding to the M
edges by starting with the first of these to the right of m, and
then continuing by the order along the horizontal axis, denoting
them by r4p2+2p1+2, ... ,r3p1+3p2+1
Finally, we can look at the diagram in figure 16, showing the set of
rectangles and transitions in this Markov partition which we now use.
A periodic symbolic sequence of allowed transitions gives as ex-
plained a periodic point in the original dynamical system. Therefore
by this diagram we can easily find other periodic orbits on the torus
that must exist for f . We will later prove that these orbits are in
fact simple, but this will require some more work. Hence, by this
diagram we prove only existence of orbits with specified rotation num-
bers. For every pair (n,m) of natural numbers, n,m 6= 0, by starting
from rp1+4p2+1, going n times around the first loop in the diagram
{rp1+4p2+1, . . . , r2p1+4p2}, then going m-1 times around the second loop
{r1, . . . , rp2} (and skipping it if m=1) and then returning through the
final sequence {r2p2+1, . . . , r3p2} to rp1+4p2+1, we get a periodic symbolic
allowed sequence, and so a new periodic orbit we denote On,m. These
Figure 16. Some of the rectangles in the Markov par-
tition, where the arrows denote allowed transitions be-
tween them
symbolic sequences are all different and hence so are the periodic orbits.
We look at a point p ∈ On,m such that p is in the rectangle rp1+4p2+1.
For the first n ·p1 iterations of p, corresponding to each time the upper
loop in the diagram appears in the symbolic sequence of p, the images
are contained in the B edges. The vertical loops are mapped under f
retaining the same ”horizontal distance” from the periodic points from
the x orbit to their left. So, p is mapped a total distance of n · q1 along
the horizontal axis under fn·p1 .
Similarly, point fnp1(p), which lies in the rectangle r1 corresponding
to D edge, is mapped a distance q2 along the horizontal axis under each
iteration of fp2 , for every occurrence of the second loop in the symbolic
sequence of p. This is because the D edges retain their distance from
the y orbit points below them. The final sequence in the symbolic
representation of p until the return to the first loop also corresponds
to the horizontal distance q1. These last points of the periodic orbit lie
in rectangles corresponding to L edges. So p is mapped a horizontal
distance of nq1 + mq2 under f
np1+mp2 . Hence, the new orbit On,m has
rotation number nq1+mq2
np1+mp2
See [12] for a proof that any two Farey neighbors span this way all
rational numbers between them, that is all rationals between q1
and q2
are of the form nq1+mq2
np1+mp2
. So we found a periodic orbit of any rational
number between the two original rotation numbers q1
and q2
Note we have found these simple periodic orbits for the Thurston-
Nielsen canonical form of the homeomorphism f we started with. It
remains to relate these periodic orbits to the periodic orbits of f itself.
Recall the following definition from [2].
A periodic point x0 ∈ S of period p for homeomorphism f0 is called
unremovable if for each given homomorphism f1 with ft : f0 ' f1 there
is a periodic point x1 of period p for f1and an arc γ : [0, 1] → S with
γ(0) = x0, γ(1) = x1 and γ(t) is a periodic point of period p for ft.
It was proven by Asimov and Franks in [2] that every periodic orbit of
a pseudo-Anosov diffeomorphism is unremovable. Thus orbits found for
the pseudo-Anosov representative exist for any other homeomorphism
in its isotopy class. This yields all these periodic orbits exist for the
original homeomorphism f as well. Thus we get theorem 3.6 for our
specific case:
Theorem 3.7. If there exists a simple pair of orbits for a homeomor-
phism f of the torus of abelian Nielsen types s and t which are Farey
neighbors, there exists a periodic orbit for f with abelian Nielsen type
equal to r for every rational number r between s and t.
4. The order relation
For any simple pair q̂1
∨ q̂2
, The orbit O1,1 out of the family of
new orbits we constructed above has rotation number equal exactly to
q1+q2
p1+p2
. This orbit corresponds to the symbolic sequence rp1+4p2+1 →
rp1+4p2+2 → ....→ r2p1+4p2 → r2p2+1 → ....→ r3p2 → rp1+4p2+1 as in the
diagram in figure 16. So we have a list of rectangles, each containing
exactly one periodic point from the new orbit O1,1. We denote the point
of O1,1 that is in a rectangle rj by oj. Graphically, assuming the first
case map, when we draw the rectangle decomposition corresponding to
the standard graph as in figure 10 we get Figure 17.
We will now show that for any simple pair q̂1
∨ q̂2
the orbit O1,1
is a simple orbit, and forms a simple pair with each periodic orbit of
the pair, that is with q̂1
and q̂2
. For the first assertion, we define
Figure 17. The intermediate orbit denoted by black
circles. The gray areas are junction, which can be
deleted, and the rectangles can be glued directly to one
another.
a family of vertical loops as follows: we choose a vertical loop that
crosses both rectangles corresponding to the m line and passes to the
right of the periodic point op1+4p2+1. Denote this loop by A. It is shown
graphically in figure 18. All its images under f until the p1st iteration
are exactly of the same form, as the rectangles are simply mapped to
the right without changing their forms. Its p1st image is the first time
it returns to the same rectangles, and is determined by the images of
the corresponding vertical edges of the graph. These images are shown
in figure 11. We use the fact f preserves orientation to determine the
relation between the image of the curve and the points of O1,1. We
denote this image by B. It is shown in figure 18.
By similar considerations, knowing the rectangles containing B in the
original picture (Figures 10 and 11) the rectangle adjacent to fp1−1(m)
on the left contains a point of the y-orbit and therefore contains a
rectangle of type L of Markov partition. This rectangle contains the
point o3p2 of the new orbit. Line A lies to the right of m and of op1+4p2+1
therefore fp1−1(A) lies to the right of o3p2 and to the right of o2p1+4p2 .
It follows that the line B = fp1(A) lies to the right of op1+4p2+1 and to
the right of o2p2+1, as shown on Figure 18. Also B can be isotoped to
the right of A relative to the points of the new periodic orbit. Next
p2 − 1 iterations of f translate A and B and whole rectangle adjacent
to m on the right to the right. We arrive at the rectangle adjacent to
Figure 18. The intermediate orbit with the family of
vertical loops
l on the left containing point o3p2 of the new orbit. The point o2p1+4p2
lies in a rectangle on the line l. The loop fp2−1(B) lies to the right of
o3p2 and to the left of o2p1+4p2 therefore the loop C = f
p2(B) lies to the
right of op1+4p2+1 and to the left of o2p2+1, as shown on Figure 18. Also
p2 iterations of f take line m to a distance p2q1 = p1q2 − 1 rectangles
to the right, which means one rectangle to the left of line m. Since A
and B are to the right of the point op1+4p2+1 in line m and since this
point moves to the leftmost point in the new periodic orbit shown on
Figure 18, loop C must be to the right of it, as in Figure 18. The point
o2p1+1 may be above or below the loop C but this does not change the
discussion bellow.
Note that if we disregard the orbits x and y of the original pair we
can isotop C to A relative the points of the new orbit. This shows that
the new orbit is a simple periodic orbit of length p1 + p2, by Lemma
2.3. Now we fill in the x orbit (the longer orbit) and consider a torus
punctured at the y orbit and the new orbit together. We have the
family of vertical loops f i(A) and the action on it is exactly as in the
condition for a simple pair as the loop C can be isotoped to A plus a
loop around y0. We choose a horizontal loop as the loop D on Figure
19. Then its image D′ is as shown on Figure 19. The image has the
required properties. The orbit y together with the new orbit form a
simple pair for the homeomorphism f .
Next we fill in the y orbit and leave punctures at the x orbit and
the new orbit. We choose the initial vertical loop A differently, as
Figure 19.
in Figure 20. This loop A is one rectangle to the right of m plus a
loop on the left. After k = p2 − 1 iterations of f it will move to
line l which is q1 rectangles to the left of m. Indeed it will move to
q1(p2 − 1) + 1 = p1q2 − q1 rectangles to the right of m which means q1
rectangles to the left. The loop fk(A) looks like the loop A and lies
to the left of the point r3p2 and to the left of the point o2p1+4p2 . Next
iteration of f takes it to a curve which looks like f(l) but lies to the
left of op1+4p2+1 and to the left of o2p2+1. Since we filled the point y0 we
can isotop this loop to a vertical loop near m, which passes to the left
of op1+4P2+1. Subsequent iterations translate it to the right and f
p1(A)
is equal to curve B on Figure 20
Next p2− 1 iterations will take B to the loop near l which lies to the
left of o2p1+4p2 . Next iteration of f takes this loop to a loop similar to
f(l), but lies to the left of o2p2+1. Since the points of y-orbit are filled
we can isotop it to the loop C on Figure 20. It can be further isotoped,
relative to the x-orbit and the new orbit, to the loop A plus a small
loop around x0 to the left of A. If we choose the same horizontal loop
as in the previous case , with the same image as before, we get the
required action of f for a simple pair consisting of the x-orbit and the
new orbit.
Now we can continue by the same analysis for each of these two
simple pairs, finding their Farey intermediate to be a simple orbit as
well that forms a simple pair with each of them, and so on. It remains
to prove the persistence of all these simple pairs under isotopies. For
this, recall The following theorem from [11].
Figure 20.
Theorem 4.1. (Hall) Let S be a closed surface and let A be a finite
subset of S. Let f be a homeomorphism of S which leaves A invari-
ant. Let p = x1, . . . , xk be a finite collection of periodic points for f
which are essential, uncollapsible, mutually non-equivalent and non-
equivalent to points of A. Then the collection p is unremovable, which
means that for every homeomorphism g isotopic to f rel A there exists
an isotopy ft rel A and paths xi(t) in S such that f0 = f , f1 = g,
xi(0) = xi, xi(t) is a periodic point of ft of period equal exactly to the
period of xi.
(This theorem is a generalization of the main result of Asimov and
Franks in [2] to several periodic orbits. In fact this generalization was
mentioned in [2] as a remark with a hint of a proof.)
Recall also that if f is pseudo-Anosov in the complement of A then
it is condensed and by [6] Lemma 1 and Theorem 2.4 each periodic
point is uncollapsible and essential and points from different orbits are
non-equivalent and points disjoint from A are not equivalent to points
of A.
Corollary 4.2. : Let T be a torus and let A be a finite subset of T .
Let f be a shear-type homeomorphism of T which is pseudo-Anosov in
the complement of A. Let g be a homeomorphism of T isotopic to f in
the complement of A. If x is a simple periodic orbit for f then there
exists a simple periodic orbit z for g with ρ(z) = ρ(x). If x, y is a
simple pair of periodic orbits for f , one or both disjoint from A, then
there exists a simple pair of periodic orbits z, w for g with ρ(z) = ρ(x)
and ρ(w) = ρ(y).
Proof. Chose points x1 and y1 from the orbits x and y . By Theorem
4.1 there exists an isotopy ft and paths x1(t) and y1(t) such that x1(0) =
x1, y1(0) = y1, x1(t) is a periodic point of ft of a fixed order p for all
t and y1(t) is a periodic point of ft of a fixed order q for all t and
y1(t) = y1(0) for all t if y(0) ∈ A. For a given t all points in the orbits
of x1(t) and y1(t) for f are distinct, they form a braid with p+q strands.
They move when t changes and their movement can be extended to an
ambient isotopy ht which is fixed on A. Then ht(f
i(x1(0)) = f
t (x1(0))
and ht(f
i(y1(0)) = f
t (y1(0)). Consider isotopy Ft = h
t ftht. We have
i(x1(0)) = f
i+1(x1(0)) and Ft(f
i(y1(0)) = f
i+1(y1(0)) so Ft is fixed
on the orbits x and y. In particular x and y form a simple pair of
periodic orbits for F1 (or x forms a simple periodic orbit for F1 if there
is no y). But F1 = h
1 gh1 so h1(x) and h1(y) form a simple pair of
periodic orbits for g.
This concludes the proof of Theorem 1.1.
5. Global analysis of the kicked accelerated particle
system
The physical system called the kicked accelerated particle consists
of particles that do not interact with one another. They are subject
to gravitation and so fall downwards, and are kicked by an electro-
magnetic field, i.e., the electro magnetic field is turned on for a very
short time once in a fixed time interval. This electromagnetic field is a
sine function of the height of the particle, hence the particles are kicked
upwards or downwards by different amounts, depending on their po-
sition at the time of a kick. For a short review of the results for this
system see [8]. Experiments of this system were conducted by the Ox-
ford group, see [17], and the system was found to show a phenomena
that is now called ”quantum accelerator modes”: as opposed to the
natural expectation that particles fall with more or less the gravita-
tional acceleration, it was found that a finite fraction of the particles
fall with constant nonzero acceleration relative to gravity, as can be
seen in Figure 21
This is a truly quantum phenomenon having no counterpart in the
classical dynamics. A theoretical explanation for this phenomenon was
given by Fishman, Guanieri and Rebuzzini in [13], and it establishes a
correspondence between accelerator modes of the physical system, and
periodic orbits of the classical map
Figure 21. Accelerator modes
Experimental Data (taken from Oberthaler, Godun, d’Arcy, Summy
and Burnett, see [17]) showing the number of atoms with specified
momentum relative to the free falling frame as the system develops in
time (the numbers on the y axis represents time by the number of
kicks, while the z coordinate is proportional to the number of atoms)
J + k̃sin(θ + J) + Ω
θ + J
mod2π(1)
Where the J coordinate corresponds to the particles momentum, and
θ to its coordinate. This map is of shear type, and the acceleration for
a periodic orbit with rotation number q
is given by
− Ω(2)
Hence, by analyzing the structure of existence of periodic orbits for
the classical map above, we would be able to find which modes should
be expected for which values of the parameters k and Ω. We remark
that actual experimental observation also requires stability of the pe-
riodic orbits. It is important to stress here that since these parameters
correspond to the kick strength and the time interval between kicks
they can be controlled in the experiments as we wish, so results ob-
tained for this system can be tested experimentally. When one plots
the numerical results describing which periods exist for different values
of k and Ω one gets an extremely complicated figure, see figure 22.
Figure 22. Tongues of periodic orbits
An exact mathematical analysis of this system is extremely compli-
cated. Perturbative methods have been used in [13] to analyze the ex-
istence of these ”tongues” of periodic orbits in the region where k → 0,
as well as giving estimates on their widths.
Look at the map f given by (1) in regions where Ω is equal q
for some rational number q
in the unit interval, and small k. For a
small enough k it can be seen both from the numerical results shown
graphically in figure 22 and from perturbative arguments that in the
above region a periodic orbit with period p exists.
For small k the periodic points of this orbit must be pretty much
equally spaced along the J axis, and we can choose (for k small enough)
a family of vertical loops that are equally spaced at distance exactly Ω
apart, and each is at distance at least, say, 3k from any of the periodic
points.
The image of a loop parameterized by Γ1(θ) =
is given by(
J0 + ksin(J0 + θ) + Ω
J0 + θ
J0 + ksin(θ
′) + Ω
and so is very
close (for small k) to another loop of the chosen family. It follows that
there exists a map f̃ isotopic to f rel the orbit which keeps this family
of curves invariant, and so, by Lemma 2.3, all the periodic orbits seen
in the tips of the tongues in Figure 22 are simple orbits.
Note the rotation number of each of these orbits is equal exactly to
the value of Ω in the tip of the tongue (k = 0) as for very small k
the J coordinate increases by an almost fixed value, close as we wish
to Ω. And, by equation (2) the rotation number q
is related to the
acceleration of the corresponding acceleration mode by
α = q
2π − Ω
So the topological meaningful numbers here are in fact also the ones
with physical significance. While Ω changes through the region in which
this periodic orbit exists, q
is of course a topological invariant and
therefore fixed. Hence the acceleration vanishes on the line with fixed
Ω in the middle of each tongue, and changes signs when one crosses
this line. This was measured experimentally in [15].
For any other point higher in the tongue which we can reach by an
isotopy along which the periodic orbit exists, we also have the orbit
is a simple orbit. We will assume, as is very natural and was checked
numerically for many cases, that the orbits remain simple throughout
the region of each tongue.
In some of the cases for which we drew a portrait of the phase space,
we found that the fact the homeomorphism is isotopic to one which is
reducible rel the periodic orbit is realized by the physical map itself, as
seen in Figure 23.
Here the phase space is truly divided into pieces. Each of the annuli
in this decomposition is mapped to another, and returns to itself with
one twist after p iterations of f . Therefore every periodic orbit must
have a period which is a multiple of p. On the other hand, when an
annulus is mapped to itself with one twist under an area preserving
map (here under fp), every rotation number in the unit interval exists
for it (here we mean the standard annulus rotation number measuring
the rotations around the annulus), and so every period exists, as for
every rational number n
there is a periodic point of order m which
rotates n times around the annulus before it returns to itself. This
yields that for such a point in the parameter space, exactly all periods
that are multiples of p exist. It is our belief that this situation is typical
Figure 23. Phase portrait for a two-orbit
Drawn for k = 1
2π and Ω = π, the two-orbit which is clearly seen is
a stable orbit with two stable neighborhoods drawn. There is another
two-orbit present, at which the arrows point, and it is the stable and
unstable manifolds for this unstable orbit which divide the phase
space into non intersecting regions which do not mix.
for the center of each tongue, that is for Ω = 2π q
. At other points,
namely in all point we have numerically checked outside the center of
the tongue, orbits of coprime lengths may exist simultaneously. We
believe that the coexisting orbits whose rotation numbers are Farey
neighbors form a simple pair together, as in the example on Figure 24,
which shows a simple pair of orbits with rotation numbers 1/3 and 1/2
found in the physical system.
This coexistence happens at a point in Figure 22 for which two
tongues intersect. We assume that the same orbit persists through-
out the tongue, and therefore we have at such a point two coexisting
simple orbits. We believe that in all points of intersecting tongues
coming from k = 0 and Ω1 =
, Ω2 =
which are Farey neighbors,
p1 > p2, the coexisting orbits form a simple pair.
Theorem 3.7 therefore implies that there are infinitely many periodic
orbits for the parameters at a region of intersection of two such tongues,
with rotation numbers equal to all rational numbers between the ones
Figure 24. A pair of coexisting orbits in the physical system
Drawn with a collection of curves on the torus and their images,
which show this is a simple pair.
of these two tongues. If we assume all these simple orbits present also
come from tongues, this yields that each rational tongue between q1
and q2
intersects each of these two tongues lower (along the k axis) than
they intersect each other. In other words, following a path from a tip of
a tongue upwards in the tongue, if it intersects a Farey neighbor tongue
we know it intersects earlier all tongues of rational numbers between
them. This determines the global structure appearing in Figure 22 of
all accelerator modes in the physical system, as Sharkovskii’s theorem
determines it for one dimensional systems.
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Tali Pinsky
Department of Mathematics
The Technion
32000 Haifa, Israel
e-mail: otali@tx.technion.ac.il
Bronislaw Wajnryb
Department of Mathematics
Rzeszow University of Technology
ul. W. Pola 2, 35-959 Rzeszow, Poland
e-mail: dwajnryb@prz.edu.pl
http://arxiv.org/abs/quant-ph/0512086
http://arxiv.org/abs/quant-ph/0512086
	Abstract
	1. Introduction
	2. Simple orbits 
	3. Simple orbit pairs
	4. The order relation
	5. Global analysis of the kicked accelerated particle system 
	References
