数理解析研究所講究録
1042 巻 1998 年 133-142

133

THE ROTATION SETS VEERSUS THE MARKOV PARTITIONS

東大数理岩井慶–郎 (KEIICHIRO IWAI)
ABSTRACT. The rotation sets of homologically trivial homeomorphisms are studied. If
a homologically trivial homeomorphism has Markov partitions, we show the relation

between this homeomorphism and the Markov partitions, give the way to calculate
the rotation set from the Markov partition, and show that if a homologically trivial
homeomorphism has Markov partitions, the rotation set is a convex polygon.

\S 1 Introduction
In this paper we show the relation between the rotation set for homeomorphisms
is the identity map and the soficonwhose associated homomorphism
system of the subshift of finite type associated with the Markov partition. We give
the way to calculate the rotation set from the Markov partition and show that therotation set is a convex polygon and explicit representation of every extremal pointof this polygon.
We will define the rotation set for homeomorphisms whose associated homomoris the identity map in Section 2, overview the theory of the shiftphism onautomorphisms in Section 3, define the Markov partitions and show some propertiesof the Markov partitions in Section 4, and show the relation betwee.n the rotation
sets and the Markov partitions in Section 5.
Our main results are:$H_{1}(M;\mathbb{Z})$$f_{*}$$f_{*}$$H_{1}(M;\mathbb{Z})$Theorem 5.2. Let $(M, f)$ be a homeomorphism whose associated homomorphismis the identity map, and suppose $(M, f)$ has a Markov partitiononof $x\in M$ is $\mathcal{I}(x)=\cdots i_{-2},i_{-1,0,}i$ i, $i2,${R. } of M. Suppose the itineraryisand the image of the 2-block mapof$f_{*}$$\mathcal{R}=$$H_{1}(M;\mathbb{Z})$$\cdots$$\mathcal{I}(x)$$\cdot$$S(\mathcal{I}(x))$$\mathcal{I}(x)$$s(\mathcal{I}(_{X))[\alpha,][]}=\cdots i-2i-1\alpha i-1,i\mathrm{o}[\alpha_{i_{0^{i}1}},][\alpha_{i},i]12\ldots$

Then the homologicd rotation set $\rho(x, f)$ of$f$is given by$p(x, f)= \sum_{i,j}P(\alpha_{i},j)[\alpha_{i,j}]$is the appearance probability of the subsequence.
$\mathcal{I}(x)=\cdots i-2,$ $i-1,i_{0},$ il,
of if

where

$P(\alpha_{i,j})$$i_{2},$$\cdots$$x$“$hR_{j}$”in the itinerary$P(\alpha_{i,j})ex\tilde{l}sts$

andTypeset by$A_{\psi t}\mathrm{S}\mathrm{I}1\pi$134homohomeomorphism whose
Theorem 5.3. Let $(M,f)$ be a
has a Markov
is the identity map andmorphismon$(f)$partitionof M. Then the rotation set Rot is a finite polygon and every
extremal point is obtained by the pointwise rotation set of some periodic point.
$a\mathit{8}SoCiated$$tran\mathit{8}itive$$\mathit{8}uppo\mathit{8}e(M, f)$$H_{1}(M;\mathbb{Z})$$f_{*}$$\mathcal{R}=\{R\}$constructed Markov partitions on the basic sets of the Axiom diffex
morphisms, so we conclude that the rotation set of Axiom diffeomorphisms with
ic set. Thus we haveis a finite polygon if is restricted on one$A$$\mathrm{B}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{n}[1]$

$A$

$f$

$f_{*}=\mathrm{i}\mathrm{d}$$\mathrm{b}\mathrm{a}s$

Corollary 5.4. For an Axiom A $diffeomo7\grave{p}$ hism urith $f_{*}=id$ , the homological
set is a finite
rotation set is a finite union of finite polygons, and the mean
polygon.
$f$$ro.tat\overline{\iota}on$$\cdot$$\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{r}\mathrm{S}\mathrm{t}_{0}\mathrm{n}[3]$constructed Markov partitions of the pseudo Anosov diffeomorphisms.

Thus we have
Corollary 5.5. For a pseudo Anosov diffeomorphism $f$ vrith $f_{*}=id_{f}$ the hommlogicalpolygon.
rotationis$.afini\dot{t}e$$\mathit{8}et$\S 2 The rotation set
Let $M$ be a closed manifold and let be a homeomorphism of $M$ whose associatedhomomorphismis the identity map. Here we define the rotation set
for and show some properties.
Suppose is a base point of . Let $p:\hat{M}arrow M$ be the maximal Abelian covering
space and $F:\hat{M}arrow\hat{M}$ be a lift of . Let $\xi\in p^{-1}(x)$ on
be a lift of on M.
denotes a curve from to on
denotes the lift of which is the closest toon .
denotes the curve given by the concatenation ofand. Sinceis the maximal Abelian, every loop on, theis mapped to a null homologous loop $p(c)$ on $M$ . Thus for every curve on
homology cl as ofis uniquely determined by the start point and the end pointof . Letbe the homology class of .
$f$$f_{*^{\mathrm{o}\mathrm{n}}}H_{1}(M;\mathbb{Z})$

$f$

$f$

$\mathrm{O}$$\hat{M}$

$f$

$b$

$a$

$\hat{M}$$a$$\hat{M}.\hat{\mathrm{O}}(a)$$\gamma(a, b)$$x$$\mathrm{O}$$\gamma(\hat{\mathrm{O}}(\xi), Fm(\xi))$$\mathcal{H}_{n}(\xi, F, \mathrm{O})$$\gamma(F^{m}(\xi), \text{\^{O}}(F^{n}(\xi)))$$(\hat{M},p)$$c$$\hat{M}$$\hat{M}$$\alpha$$p(\alpha)$$\mathrm{s}$$\alpha$$[\alpha]$$\alpha$[Figure 2.1]

Proposition 2.1. For $\xi,\eta\in p^{-1}(X),$Let$p(\mathcal{H}n(\xi, F, \mathrm{O}))=p(\mathcal{H}_{n}(\eta, F, 0))$denote the loop on $M$ which is the image of of the curveanddenote the homology class of $h(x, f, F’, 0)$ .

$h_{n}(x, f, F, \mathrm{O})$$\mathcal{H}_{n}(\xi, F, \mathrm{O})$on$\hat{M}$$p$$[h_{n}(x, f, F, \mathrm{O})]$

Proposition 2.2.
$[h_{n}(x,f, F, \mathrm{O})]=\sum_{i=0}^{n-1}[h1(\dot{F}(X), f,F, 0)]$

135
[Figure 2.2]

Proposition 2.3. Let $F$ andanddoes not depend on .

$G$be lifls of $f$ . Then the difference between the element, andoffor some element$[h_{n}(x, f, F, \mathrm{O})]$$H_{1}(M;\mathbb{Z})$$\alpha(h)$$[h_{n}(x, f, G, \mathrm{O})]i\mathit{8}n\alpha(h)$$x$$\alpha(h)$Proposition 2.4. Let A and are the different base points on M. Then.
is bounded element of$\mathrm{B}$$[h_{n}\langle x, f, F,\mathrm{A})]-\mathrm{I}$$H_{1}(M;\mathbb{Z})$$[h_{n}(X, f, F, \mathrm{B})]$

Definition 2.5. The (homological) pointurise rotation set $p(x, f, F)$ of with respect
to and $F$ is defined as
$x$

$f$

$\rho(x, f, F)=\lim_{arrow n\infty}\frac{[h_{n}(X,f,F,0)]}{n}$if limit exists.
.
Note that $\rho(x, f, F)$ is an element of$F$
Suppose and are lifts of then there is an element of the covering transformation $D(\hat{M},p, M)$ which satisfies $G=h\circ F$ . Let $\alpha(F, G)$ be an element of
which is defined by . Then the difference between $p(x, f, F)$ and $p(x, f, G)$ is equal
to $\alpha(F, G)$ and $\alpha(F, G)$ is not depend on .
$f,$ $F)|x\in M$ and $p(x,$ $f,$ $F)$ exists} of the
The set Rot $(f, F)=$ {pointwise rotation set $\rho(x, f, F)$ is called the rotation set of .
$H_{1}(M;\mathrm{R})$

$G$

$h$

$f$

$H_{1}(M;\mathbb{Z})$

$h$

$x$

$\rho(x,$$\mathrm{h}_{\mathrm{o}\mathrm{m}\mathrm{o}}10\dot{g}\mathrm{c}\mathrm{a}1$$f$Remark. Whenof Franks [6].

$f$is homotopic to the identity map, this definition agrees with that

Proposition 2.6. $\rho(x, f^{N}, F^{N})=N\cdot p(x, f, F)$

Let. us consider the mean rotation set with respect to the measure$\mu$

on $M$ . Firstly

let us recall the following theorem.

Theorem (Birkhoff’s ergodic theorem). Let be a measure on $M$ and suppose: $(M, \mu)arrow(M, \mu)$ is -preserving and $h\in L^{1}(\mu)$ . Then
$\mu$

$f$

$\mu$

$\frac{1}{n}\sum_{i=0}^{n-1}h(f^{i}(x))$

converges
If$f$$\mathrm{a}.\mathrm{e}$. to a function$h^{*}\in L^{1}(\mu)$

prservae the measure

$\mu$

.

on $M$ , ffom the Birkhoff’s ergodic theorem, the limit$\lim_{narrow\infty}\frac{[h_{n}(X,f,F,0)]}{n}=narrow\lim\infty\frac{1}{n}\sum_{i=0}^{n}[h1(f^{i}(x),f, F, \mathrm{O}-1)]$

136
exists for almost every with respect to .
denotes an $f$-invariant measure on $M$ . Let $p_{\mu}(f, F)$
In the following
be the mean rotation set defined as
$x$

$\mu$

$\mathrm{a}\mathrm{r}_{\mathrm{o}}\mathrm{p}\mathrm{u}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s},$$\mu$$p_{\mu}(f, F)= \int_{M}\rho(x, f, F)d\mu$

Proposition 2.7 (The ergodic theorem). The mean rotation set satisfies theequality.

$f_{oll_{\mathit{0}}}u\dot{n}ng$$p_{\mu}(f, F)= \int_{M}[h_{1}(X, f, F, 0)]d\mu$Lemma 2.8. Let $M$ be a compact manifold and let and be homeomorphisms of$M$ whose associated $homomorphi\mathit{8}mf*onH_{1}(M, \mathbb{Z})$ is themap. We also
be a lifl ofmeasure on M. Let
and have the common
to. Then the rotation set
the compositionof the compositionof and is equal to the sum $\rho_{\mu}(f, F)+p_{\mu}(g, G)$ of the rotation set $p_{\mu}(f, F)$translation ambiguity.
of urithof and that
$f$

$g$

$ld\sim ent_{\tilde{l}t}y$$\mathit{8}upposef$$g\circ f$$f$$g\mathrm{o}f$$\overline{g\circ f}$$invan\overline{a}nt$

$g$

$\mu$

$p_{\mu}(g\circ f,\overline{g\circ f})$$\hat{M}$$g$$\rho_{\mu}(g, G)$

$f$

$g$

$H_{1}(M;\mathbb{Z})$\S 3 Shift Automorphisms
Let be a positive integer and $[k]$ be the set of numbers $\{1, 2, \cdots, k\}$ with the.
be the product space
discrete topology. $[k]$ is called the alphabet set. Let, whereis an infinite sequence $a=\cdots a_{-2},$ $a_{-1,0,1}aa,$ $a2,$
Then an element of$[k]=\{1,2,
-.
,
k\}$
inducesontopology
product.
The
is contained in
every
$k$

$[k]^{\mathbb{Z}}$$\Sigma(k)$$\cdots$$\Sigma(k)$$\Sigma(k)$$a_{n}$

the metric $d(a, b)= \sum_{n=-\infty}^{\infty}2^{-}21n|-1\delta(na, b)$, where $\delta_{n}(a, b)=0$ if $a_{n}=b_{n}$ otherwise
is compact, totally. Thenis homeomorphic to the Cantor set.
thus
Let the shift a be the homeomorphism on$\delta_{n}(a, b)=1$$\Sigma(k)$$\mathrm{d}\mathrm{i}_{\mathrm{S}\mathrm{C}\mathrm{o}}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{C}.\mathrm{t}\mathrm{e}\mathrm{d}$and has no isolated points,$\Sigma(k)$) defined as $(\sigma(a))_{n}=a_{n+1}$ where. The shift moves thedenotes the n-th digit of the infinite sequence
sequence one place to the left.
Next we define the subshift of finite type.
$\Sigma\langle k$$\sigma(a)$$(\sigma(a))_{n}$Definition 3.1. Let $A=(A_{ij})$ be aasofsubspace$\Sigma_{A}$$k\cross k$matrix of 0-1 entries. We define the$\Sigma(k)$$\Sigma_{A}=${ $a\in\Sigma(k)|A_{aa}::+1=1$ for every }$i$is also. The restriction of a on
is a closed -invariant subspace ofThentype and the matrix is
a subshift ofwritten as . We call the pairan admissible sequence.
iscalled the transition matrix. Every member ofthen theIf for every and , there is a positive integer $n(ij)$ such thatmatrix is called transitive.
$\Sigma_{A}$$\Sigma(k)$$\sigma$$\Sigma_{A}$

$i$

$A$

$j$

$A$

$fim\tilde{t}e$$(\Sigma_{A}, \sigma)$$\sigma$$\Sigma_{A}$$\mathrm{c}\mathrm{a}\mathrm{U}\mathrm{e}\mathrm{d}$$A_{ij}^{n(ij)}\neq 0$

137
For

$\{c_{i}, c_{i+1}, \cdots,\alpha_{+j}\}\subset[k]$, the set$C(c_{i}, \mathrm{G}+1, \cdots, c_{\check{\iota}+j})=\{a\in\Sigma_{A}|a_{k}=c_{k},$$i\leq k\leq i+_{\acute{J}\}}$

is called a cylinder.
The subshift of finite type defines the oriented graph whose vertices are $[k]$ andwhose oriented edges are given by the matrix $A$ as follows:, we attach the oriented path $\gamma(C(c_{0}, c_{1}, \cdots , c_{j}))=$For a cylinder
to this sequence. Inver.sely, we can determine the$c_{0}arrow c_{1}arrow c_{2}arrow\cdotsarrow c_{j}$ ofcylinder $C$ from the finite path on .
An oriented path $a_{0}arrow a_{1}arrow a_{2}arrow\cdotsarrow a_{m}$ on is called a simple loop whenfor every $0\leq i<j\leq m-1$ .
the path satisfies $a_{0}=a_{m}$ and, we define elements
Let $V$ be the vector space hulled by $[k]$ and for in$\Gamma$$C(c_{0}, c_{1}, \cdots , c_{j})$$\Gamma$$\Gamma$$\Gamma$$a_{i}\neq a_{j}$$(\Sigma_{A}, \sigma)$$a$$v_{n}(a)$and $v(a)$ of$V$

as follows:$v_{n}(a)= \frac{a_{0}+\cdots+a_{n}}{n+1}$$v(a)=narrow\infty 1\dot{\mathrm{m}}1v(na)$is periodic with period,$a0,$Suppose $a=\cdots a_{0},$. Let be a finite loop on given by $l_{i}=(a_{0}arrow a_{1}arrow\cdotsarrow a_{p-1}arrow a_{0})$ . Then$a_{1},$$p$

if limit exists$\cdots$$a_{\mathrm{p}-1},$$a_{1},$$\cdots,$$a_{p-1},$$\cdots\in\Sigma_{A}$$\Gamma$$l_{i}$

periodic point of$(\Sigma_{A}, \sigma)$corresponds to some loop of F.

be a subshifl offinite type of the transitive transition matrix
Lemma 3.2. Let
urith alphabet set $[k]=\{1,2, \cdots , k\}$ and be the graph given by the, .
denotes the $ba\eta center$ in
A. Let be the vector space hulled by 1, 2,
corresponding to the rimple loop of F.
, we define the vector
For $eve\eta admis\dot{\Re}ble$ sequence $a=\cdots a_{-1}a0^{a_{1}}\cdots$ ofand in as$A$$\Sigma_{A}$$tr.an\mathit{8}itionmat7\dot{\mathrm{B}}X$$\Gamma$

$V$

$\{$

$\cdots$

$k\}$

$V$

$g_{i}$$l_{i}$$\Sigma_{A}$$v_{n}^{+},v_{n},$ $v^{+}$

$v$

$V$

$v_{n}^{+}(a)= \frac{1}{n+1}(a_{0}+a_{1}+\cdots+a_{n-1}+a_{n})$$v^{+}(a)= \lim_{narrow\infty}v_{n}(+a)$

if limit$ex\tilde{\iota}sts$.

$v_{n}(a)= \frac{1}{2n+1}(a-n+a_{-}n+1+\cdots+a-1+a0+a1+\cdots+a_{n-1}+a_{n})-$$v(a)= \lim_{narrow\infty}v_{n}(a)$if limit exists

Then the set $P^{+}=\{v^{+}(a)|a\in\Sigma_{A}\}$ agrees vrith $P=\{v(a)|a\in\Sigma_{A}\}$ , and each ofisthem is equal to the closure of the convex hullof G. Moreover,a convex finite polygon.
$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{V}(c)$$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{V}(c_{7})$138\S 4 Markov partitions
In this section, we define a Markov partition and show some properties.
Let $M$ be a compact manifold and : $Marrow M$ be a homeomorphism of $M$ .
$f$be a subset of $M$ . An (at most countable) partition ofDefinition 4.1. Let$M$ is called a topological generator
if the following conditions are satisfied:for$M_{0}$$\alpha$$M_{0}$(1) The union$A\in\alpha$$\bigcup_{A\in\alpha}\mathrm{h}\mathrm{t}A$of the interior$\mathrm{I}\mathrm{n}\mathrm{t}A$is dense in.nnd$M\mathrm{a}$$\mathrm{I}\mathrm{n}\mathrm{t}A\neq\emptyset$

for every.

(2) if $x\in M_{0}$ and every sequence$A_{i_{k}}\in\alpha$satisfies that$\cap nf^{k}(\mathrm{I}\mathrm{n}\mathrm{t}Aik)$$x\in\cap$$n\in \mathbb{Z}k=-n$then$\cap nf^{k}(\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{A}_{k})$$\{x\}=\cap$$n\in \mathbb{Z}k=-n$

where$\overline{A}$is the closure of$A$.

$(A_{1}, A_{2}, \cdots , A_{N})$ for $M$ satDefinition 4.2. A finite topological generator
for every is called a Markov partition for $M$ if the foliowing
isfying
conditions are. satisfied:$\alpha=$$i$$A_{i}\subset\overline{\mathrm{I}\mathrm{n}\mathrm{t}A_{i}}$(1) (Local product structure) For every 4, there exist compact spaces
and a topological isomorphism$\varphi_{i}$such that for every$x\in \mathrm{I}\mathrm{n}\mathrm{t}\mathrm{A}$

with

:

$E_{i}$

and$F_{i}$,$\overline{\mathrm{A}}arrow E_{i}\mathrm{x}F_{i}$$f(x)\in \mathrm{I}\mathrm{n}\mathrm{t}A_{j}$for some $1\leq j\leq N$ ,$f(E_{i}(x)\cap \mathrm{I}\mathrm{n}\mathrm{t}A_{i})\supset E_{j}(f(x))\mathrm{n}\mathrm{I}\mathrm{n}\mathrm{t}Aj$$f^{-1}(F_{j(f}(x))\cap \mathrm{I}\mathrm{n}\mathrm{t}A_{j})\supset F_{i}(x)\cap \mathrm{I}\mathrm{n}\mathrm{t}A_{i}$where $E_{i}(x)=\{y\in E_{i}|\varphi_{i}(y)\in E_{i}\cross\{x_{2}\}\}$ and $F_{i}(x)=\{y\in F_{i}|\varphi_{i}(y)\in$$\{x_{1}\}\cross F_{i}\}$$.(2)$.

(Boundary condition) There exists a decomposition$M \backslash \bigcup_{i=1}^{N}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{A}=B^{+}\cup B^{-}$(not necessanily disjoint union) such that$f(B^{+})\subset B^{+}$,$f^{-1}(B^{-})\subset B-$

139

de.finae the transition matrix

When $(M, f)$ has a Markov partition, thismatrix which is defined asofif
1$t_{ij}=\{$
otherwise$l\cross l$$T=(t_{ij})$$f(\mathrm{I}\mathrm{n}\mathrm{t}R)\cap R_{j}\neq\emptyset$$0$.
This transition matrix $T$ defines the subshift of finite typeof is an element of
The itinerary$f^{k}(x)$$M$, define $i_{k}=l$ or
and
is on
if $f^{k}(x)\in R_{l}$ for every in . When$i_{k}=m$ properly.
.
Let $M$ be a manifold and be a homeomorphism on $M$ . Suppose $f$ on $M$ has
be a Markov partition of $M$ . Then we
Markov partitions and let
and $f$ on $M$
can define the semiconjugacy of on the subshift of finite typeas follows;,is, we can find in $M$ whose itinerary
For every awhere $i_{k}=j$ if $f^{k}(x)\in R_{j}$ . denotes this map. Since every in $M$ has its itinerary,
is clearly surjective. We have the two representation of a rational decimal which
the direct calculations [16].
makes finite to 1. The continuity of is given
Thus we have the folowing semiconjugacy$(\Sigma_{T}, \sigma)$$x$$\mathcal{I}(x)=\cdots i_{-1}i_{01}ii_{2}\cdots$$\Sigma_{A}\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}.\mathfrak{B}i_{k}=l$$\partial R_{m}$$\partial R_{l}$

$x$

$f$

$\mathcal{R}=\{R_{1}, \cdots , R_{l}\}$$(\Sigma_{T}, \sigma)$$\sigma$$\mathrm{i}\dot{\mathrm{n}}\Sigma_{T}$$\mathcal{I}(x)$$x$$a=\cdots i_{-1}i_{\mathit{0}^{i}1}i_{2}\cdots$$x$$\theta$$\theta$$\mathrm{h}\mathrm{o}\mathrm{m}$$\theta$$\theta$$\Sigma_{T}rightarrow\sigma\Sigma_{T}$$\downarrow\theta$$\theta\downarrow$$Mrightarrow fM$

where$\theta$

is at most$k^{2}$\S 5 The rotation set

to 1.
$\mathrm{v}.\mathrm{s}$. the$\mathrm{M}\mathrm{a}\mathrm{r}\mathrm{l}\infty \mathrm{v}$partition

Here we show the relation between the rotation set $p(x, f)$ and the Markov partitions.
In this section we suppose $(M, f)$ has a Markov partition
denotes this subshift of$M:(M,
f,\mathcal{R})$ induces a subshift of finite type andoffinite type.
Let $0$ be a base point on $M$ and $(\hat{M},p, M)$ be the maximal Abelian coveing space.
to every in $M$ . For the Markov partition, take a
We attach the itinerary
and let this point denote the representative pointofpoint in the interiorbeand letbe a lifted point ofof the rectangle $R$ . Let, there is athe lifted rectangle of $R$ which contains . Suppose Int
be a point in. Letsuch thatofliftand by a. We connect
be the representative point
andondenotes this simple curve. Then we have the curve
simple curve andwhich defines the loop
andby the concatenation of.
denote the homology class of the loop
on $M$ . Let$\mathcal{R}=\{R_{1}, R_{2}, \cdots , R_{l}\}$$(\Sigma_{A}.’\sigma)$$x$$\mathcal{I}(x)\in\Sigma_{A}$$\mathrm{I}\mathrm{n}\mathrm{t}R\dot{.}$$R\dot{.}$$x_{i}$$\hat{R}\dot{.}$$\xi_{i}\in p^{-1}(x_{i})$$x_{i}\in \mathrm{I}\mathrm{n}\mathrm{t}R$$f(R)\cap \mathrm{I}\mathrm{n}\mathrm{t}R_{j}\neq\emptyset$$\xi_{i}$$\hat{R}_{j}$$\mathrm{I}\mathrm{n}\mathrm{t}F(\hat{R}\dot{.})\cap \mathrm{I}\mathrm{n}\mathrm{t}\hat{R}_{j}$$\mathrm{I}\mathrm{n}\mathrm{t}F(\hat{h})\cap \mathrm{I}\mathrm{n}\mathrm{t}\hat{R}_{j}\neq\emptyset$$R_{j}$$\xi_{ij}$$\mathrm{o}\mathrm{f}\cdot\hat{R}_{j}$$\xi_{j}\in p^{-1}(x_{j})$$F(\xi_{i})$$\xi_{j}$$\hat{M}$$\mathrm{t}_{j}$$g_{ij}$$\gamma(\hat{\mathrm{O}}(\xi_{i}), F(\xi i)),$$\alpha_{ij}$$\gamma(\xi_{j},\hat{\mathrm{O}}(\xi_{j}))$$g_{ij}$$[\alpha_{ij}]\in H_{1}(M;\mathbb{Z})$$\alpha_{ij}$[figure 5.1]

140of the subshift of finite typeDefinition 5.1. 2-block map $S$ :is a homomorphism defined as follows.
to.
.
.
. For every pairis an element of$a2,$
Suppose $a=$
. Thenof, there is an element,of$S(a)$ is defined by$\Sigma_{A}arrow(H_{1}(M;\mathbb{Z}))\mathrm{z}$$(H_{1}(M;\mathbb{Z}))\mathbb{Z}$$\Sigma_{A}$$a_{-2,-1,\mathrm{O}}aa,$$i\in \mathbb{Z}$$(a_{i}, a_{i+1})$$\Sigma_{A}$$\cdots$$a_{1},$$a\in\Sigma_{A}$$[\alpha_{aa}]:,:+1$$H_{1}(M;\mathbb{Z})$$s(a)=\cdots 1^{\alpha}a_{-2},a-1]1^{\alpha}a_{-1},ao][\alpha a\mathrm{o}^{a},1][\alpha]a_{1},a_{2}\ldots$

Note that$(S(\Sigma_{A}), \sigma)$

is a typical example of the sofic system.[21]Theorem 5.2. Let $(M, f)$ be a homeomorphism whose associated $h_{omo}mo7phismf*$
is the identity map, and $\mathit{8}uppose(M, f)$ has a Markov partitionon$i0,$ il,$\{R\}$ of M. Suppose the itineraryof $x\in M$ isand the image of the 2-block mapof$\prime \mathcal{R}=$$H_{1}(M;\mathbb{Z})$$\mathcal{I}(x)=\cdots i_{-2},$ $i_{-}1,$$\mathcal{I}(x\rangle$$S(\mathcal{I}(x))$$i_{2},$$\cdots$$\mathcal{I}(x)i\mathit{8}$$S(\mathcal{I}(x))=\cdots[\alpha_{i_{-2},i_{-}}]1[\alpha_{ii\mathrm{o}}-1,][\alpha i\mathrm{O},i1][\alpha i1,i2]\cdots$

Then the homological rotation set $p(x, f)$ of$f$is given by$\rho(x, f)=\sum_{i,j}P(\alpha_{i},j)[\alpha_{i,j}]$

where$P(\alpha_{i,j})$$\mathcal{I}(x)=\cdots i_{-2},$is the appearance probability of the subsequence $ilRR_{j}$ ” in the itirierary$i_{-1,0}i,$ il, $i2,$
exist8.
of if$\cdots$$x$$P(\alpha_{i,j})$Noting that a homomorphic image of the convex finite polygon is also the convex
finite polygon, from the Lemma 3.2, we have
Theorem 5.3. Let $(M, f)$ be a transitive homeomorphism whose associated homois the identity map and $\mathit{8}uppose(M, f)$ has a Markov
morphismonpartitionof M. Then the rotation set Rot $(f)$ is a finite polygon and every
extremal point is obtained by the pointurise rotation set of some periodic point.
$f_{*}$$H_{1}(M;\mathbb{Z})$$\mathcal{R}=\{R\}$constructed Markov partitions on the basic sets of the Axiom diffeomorphisms, so we conclude that the rotation set of Axiom diffeomorphisms withis a finite polygon if is restricted on one basic set. Thus we have$A$$\mathrm{B}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{n}[1]$

$A$

$f$

$f_{*}=\mathrm{i}\mathrm{d}$Corollary 5.4. For an Axiom A $diffeomo7phiSmfu\tilde{n}thf_{*}=id$, the homologicd
is a finite union of finite $polygons_{J}$ and the mean rotation set is a finite
rotationpolygon.
$\mathit{8}et$$\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{o}\mathrm{e}\mathrm{t}\mathrm{o}\mathrm{n}[3]$Thus we have.

constructed Markov partitions of the pseudo Anosov diffeomorphisms.

141
Corollary 5.5. For a pseudo Anosov diffeomorphism $f$ vrilh $f_{*}=id$, the homologicalrotation set is a finite polygon.
Acknowledgements

The author would like to thank his advisor, Koichi Yano, for his encouragement
and helpful suggestions.

REFERENCES
1. Bowen, R., Markov partitions for$A\acute{m}om$

A diffeomorphtsms, Amer. J. Math. 92 (1970), 725-

747.
of $Su\tau faC$es afler Nielsen and Thurston, Cam2. Casson, A. J. and Bleiler, .S- A.,
bridge, 1988.
3. Fathi, A. Laudenbach, F. and Poe’naru, V-, ltuvaux de Thurston sur les surfaces, Ast\’erisque$Aut_{\mathit{0}}mor\mathrm{p}hi_{S}.ms$

66-67 (1971).
, Invent. Math.
and periodic paints of annulus
4. Franks, J., Geodesics on
108 (1992), 403-418.
5. Handel, M., The rotation set of a homeomorphism of the annulus is closed, Comm. Math. Phis.
.
127 (1990),
Rotat\’ion
are close4 Math. Proc. Camb. Phil. Soc. 89 (1981), 107-111.
sets
R.,
Ito,
6.
MaeKeyand Guckenheimer J., Resonance regions for families of $tor[B$ maps,
R.
S.
S.,
Kim,
7.
Nonlinearity 2 (1989), 391-404.
8. Kwapisz, J., Every convex polygon urith rational vertices is a rotation set, Ergod. Th. and Dyn.
Sys. 12 (1992), 333-339.
tofus. 9. Llibre, J. and Mackey, R. S., Rotation vectors and ent ropy for homeomorphisms of(1991),
115-128.
11
Sys.
Dyn.
to ident\’ity, Ergod. Th. and10. Matsumoto, S., RDtation sets for su’faCe homeomorphisms, preprint.
ian, K., Rotation sets for maps of tori, J. Lond. Math. Soc. 40 (1989),
11. Misiurewicz, M. and$S^{2}$$homeomof\mathrm{P}hl\vee s\tau \mathfrak{U}$$33\mathrm{g}_{-}349$$\tilde{l}sotopic$

490-506.

Ziem.

12. Misiurewicz, M. and Ziemian, K., Rotation sets and ergod\’ic measures for tortts homeomorphisms, Fund. Math. 137 (1991), 45-52.
13. Newhouse, S., Palis, J. and Takens, F., Bifivrcations and stabdit.y of famikes of diffeomo rphisms,
Publ. I. H. E. S. 57 (1983), 5-71.
14. Pollicott, M., Rotation sets for homeomorphisms and hwmology, Trans.A.M.S. 331 (1992).
15. Sasaki, Y., Pseudo Anosov diffeomorphisms on the orientable surface of genus \’infin\’ity (Japanese), Master’s thesis, Univ. Tokyo (1995).
16. Schwarzman, S., Asymptotic cycles, Annals of Math. 66 (1957), 270-284.
17. Shub, M., Grobal Stab\’ility of Dynamical Systems, Springer, 198618. Sinai, Ya. G., Markov partitions and C d\’iffeomorphtsms, Functional Annal. Appl. 2 (1968), 64.
19. Sinai, Ya. G., Construc tion of Markov partitions, Fhnctional Annal. Appl. 2 (1968), 70.
20. Coven, E. M. and Paul, M. E., Sofic systems, Israel J. Math. 20 (1975), 165-177.
21. Weiss, B., Subsh\’ifts of finite twe and sofic systems, Monatsh. Math. 77 (1973), 462-474.
22. Smale, S., Different\’iable Dynamical Systems, Bull. Amer. Math. Soc. 73 (1967), 747-817.
23. Denker, M. Grillenberger, C. and Sigmund, K., Lecture Notes in Math. 527 (1976), Springer.

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