# Cantor Limit Set of a Topological Transformation Group on S1

Authors: Keying Guan; Zuming Chen
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

---

## Abstract

The limit set of a topological transformation group onS1 generated by two generators is proved to be totally disconnected (or thin) and perfect if the conditions (i–v) are satisfied. A concrete application to a Doubly Periodic Riccati equation is given.


---

## Body

## 1. Introduction

The conceptionlimit set of a transformation group plays an important role in both theory and application of modern mathematics.Assume thatΓ is a transformation group (or semigroup) formed with some continuous self-mapping on a Hausdorff space X. For any x0∈X, the set (1.1)Γ(x0)={γ(x0)∣γ∈Γ}
is called an orbit through x0 under the action of Γ. For any subset A of X, let (1.2)Γ(A)=⋃x∈AΓ(x).
A subset A of X is called a Γ-invariant set if (1.3)Γ(A)=A.
A subset A of X is called the least invariant set of Γ if it is a nonempty closed invariant set, in which there is not any nonempty closed proper subset which is Γ-invariant. Based on the continuity of the elements in Γ, it is easy to obtain the following proposition.Proposition 1.1.
LetA be a least invariant set of  Γ. For any  x0∈X, if there is such a point  a∈A that
(1.4)a∈Γ(x0)¯,
then
(1.5)A⊂Γ(x0)¯.For anyx0∈X, a least invariant set A of Γ is called a limit set of the orbit Γ(x0) if (1.6)A⊂Γ(x0)¯.
It is easy to prove the following proposition.Proposition 1.2.
If the topology spaceX is compact, then under the action of the transformation group (or semigroup)  Γ, any least invariant set is perfect if it is not finite.Therefore, it is quite possible that a limit setA under the action of Γ may be totally disconnected and perfect (a Cantor set), and that A may be with fractal structure when some measure is attached.It is an important subject in the modern nonlinear science to study the exact structure of the limit set for a given nonlinear system, especially, to determine if the limit set is a totally disconnected and perfect set (for simple, called a Cantor set). However, it is usually not an easy task to do so, because of the strong nonlinearity and nonintegrability of the system. So it is necessary to explore the conditions for the existence of Cantor-type limit set.As an example, a traditional dynamical system is a continuous (or discrete) group or a semigroupϕt (or ϕn), acting on a manifold M. The related group or semigroup is usually generated by a single generator. Both the ω-limit set and α-limit set of an orbit through a point x0 are limit sets of the corresponding group by the present definition [1]. If a least invariant set or a limit set of a dynamical system is Cantor type, then the related complicated motion is described as deterministic chaos. This kind of complicated motion has been considered widely. And some methods for determining if the least invariant set of a dynamical system is Cantor type, such as Melnikov function method, have been well developed.In the study of structures of high-dimensional leaves of a foliation in modern geometry theory, or concretely, for a differential equation in the complex domain, in the study of structures of the foliation formed with the solution manifolds (Riemann surfaces) as its leaves in phase space, it is in need to study the correspondingmonodromy grouporholonomy group, which is a kind of representative of the fundamental group of the foliation on its leaf [2–4]. Different to the dynamic system, the monodromy group is usually generated by several generators. Clearly, a complicated limit set of a monodromy group should reflect the complicated structure of the corresponding foliation and its leaves.Besides the interest in geometry, the structure of the least invariant set of the monodromy group usually relates the integrability of the corresponding ordinary differential equation. In different theories on the integrability of ordinary differential equations, such as the analytical theory [3, 5, 6], Lie group theory [7, 8], differential Galois theory [9–11], and so forth, it is commonly implied that, for almost all of the integrable ordinary differential equations, the least invariant sets of foliations in complex domain should have a simple structure, or precisely, the corresponding monodromy groups should be solvable [3, 10].Concretely, for a second-order linear homogeneous ordinary equation with rational function coefficients, it is well known that every second-order linear homogeneous ordinary equation is corresponding to a Riccati equation [12]; if it is integrable in quadratures, then every least invariant set (including the limit set) of the monodromy group for the corresponding Riccati equation should be a finite set, and each finite least invariant set is corresponding to an algebraic curve solution of the Riccati equation. Otherwise, if a limit set of the monodromy group is not finite, especially if it is of Cantor type, then the Riccati equation and its corresponding second-order linear homogeneous ordinary equation is clearly not integrable in quadratures.In order to investigate a Riccati equation with more complicated coefficients, Guan et al. studied a concrete doubly periodic Riccati equation with a Weierstrass elliptic function coefficient [13–18]; Guan considered its monodromy group and proposed roughly a method to check if its limit set is Cantor type based on numerical result.In the present paper, this method is improved into a theorem in a more exact and more general form in Section2. Combining the result in [19], the critical parameter for the existence of Cantor limit set is given exactly in Section 3.


## 2. The Theorem and Proof

Theorem 2.1 ..
LetΓ be a transformation group on the 1-dimensional sphere,
(2.1)S1=RZ.G is generated by two generators a and b, where a and b are both homeomorphic transformations onto S1. The least invariant set of Γ is totally disconnected and perfect and is also the unique limit set of any orbit Γ(x) if all of the following conditions (i–v) are satisfied. (i)
Botha and b have exactly two fixed points in S1,α1,α2, and β1,β2, respectively, that is,(2.2)a(αi)=αi,b(βi)=βi,i=1,2,
where β1 is located in the inside of one of the two arcs of S1 separated by α1 and α2, while β2 is located in the inside of the other one. (ii)
The arcsα1β1α2̂ and α1β2α2̂ are both invariant sets of a, that is,(2.3)a:α1β1α2̂⟼α1β1α2̂,a:α1β2α2̂⟼α1β2α2̂,
and the arcs β1α1β2̂ and β1α2β2̂ are both invariant sets of b.(iii)
For bothi=1 and i=2,(2.4)lim⁡n→∞an(βi)=α2,lim⁡n→∞a-n(βi)=α1,lim⁡n→∞bn(αi)=β2,lim⁡n→∞b-n(αi)=β1.(iv)
The commutator ofa and b,(2.5)h=b-1a-1ba(∈Γ),
has exactly two different fixed points ξ1 and η1 located in the inside of the arc α1β1̂, that is,
(2.6)ξ1,η1∈(α1β1)̂.
where ξ1 is closer to α1 than η1. In this paper, (α1β1)̂ represents the open arc without two ends α1 and β1, and [α1β1]̂ represent, the closed arc with the two ends α1 and β1,(v)
Under the  actions ofa and b, the points ξ1 and η1 are transformed into other three arcs different to α1β1̂:(2.7)ξ2=a(ξ1),η2=a(η1),ξ2,η2∈(β1α2)̂,ξ3=b(ξ2),η3=b(η2),ξ3,η3∈(β2α2)̂,ξ4=b(ξ1),η4=b(η1),ξ4,η4∈(α1β2)̂.Proof.
Let(2.8)I0=(ξ1η1)̂.
By the homeomorphic property of transformations and conditions (ii) and (v), we can see
(2.9)(ξ2η2)̂=a(I0),(ξ3η3)̂=b(a(I0)),(ξ4η4)̂=b(I0)(=a-1(b(a(I0)))).
Let
(2.10)O(0)=I0⋃a(I0)⋃b(a(I0))⋃b(I0),
and let
(2.11)F(0)=S1∖O(0);
then the following facts and conclusions can be derived step by step.Fact 1.
The closed setF(0) is formed with following four separated closed arcs:
(2.12)Fα1=[η4α1ξ1]̂,Fα2=[η2α2ξ3]̂,Fβ1=[η1β1ξ2]̂,Fβ2=[η3β2ξ4]̂.Fact 2.
Obviously,(2.13)a(Fα1)=Fβ2⋃Fβ1⋃Fα1⋃I0⋃b(I0),a-1(Fα2)=Fβ2⋃Fβ1⋃Fα2⋃a(I0)⋃b(a(I0)),b(Fβ1)=Fα2⋃Fα1⋃Fβ1⋃I0⋃a(I0),b-1(Fβ2)=Fα2⋃Fα1⋃Fβ2⋃b(I0)⋃b(a(I0)).Fact 3.
From (i), (ii), and (iii), it follows that(2.14)lim⁡n→∞an(x)=α2,lim⁡n→∞a-n(x)=α1,∀x∈S1∖{α1,α2},
and that
(2.15)lim⁡n→∞bn(x)=β2,lim⁡n→∞b-n(x)=β1,∀x∈S1∖{β1,β2}.Fact 4.
Clearly, the commutatorh of a and b is also a homeomorphic transformation. From the condition (iv), it follows that, either
(2.16)lim⁡n→-∞hn(x)=ξ1,lim⁡n→+∞hn(x)=η1
or
(2.17)lim⁡n→-∞hn(x)=η1,lim⁡n→+∞hn(x)=ξ1
if
(2.18)x∈S1∖{ξ1,η1}.Fact 5.
LetZ0 represent the set of all nonzero integers, and let
(2.19)Fα=Fα1⋃Fα2,Fβ=Fβ1⋃Fβ2;
then the family of transformations {am}m∈Z0 changes Fβ into a family of separated closed arcs in Fα, which are condensed at α1 and α2, and the family of transformation {bm}m∈Z0 changes Fα into a family of separated closed arcs in Fβ with β1 and β2 as their condensation points.Fact 6.
Let(2.20)Fα(1)=⋃m∈Z0am(Fβ)⊂Fα,Fβ(1)=⋃m∈Z0bm(Fα)⊂Fβ,
and let
(2.21)O(1)=⋃m∈Z[am(O(0))⋃bm(O(0))],F(1)=Fα(1)¯⋃Fβ(1)¯;
then it follows that
(2.22)F(1)=S1∖O(1).
For an integern greater than 1, we may inductively let
(2.23)Fα(n)=⋃m∈Z0am(Fβ(n-1))⊂Fα(n-1),Fβ(n)=⋃m∈Z0am(Fα(n-1))⊂Fβ(n-1),O(n)=⋃m∈Z[am(O(n-1))⋃bm(O(n-1))]⊃O(n-1),F(n)=Fα(n)¯⋃Fβ(n)¯(=S1∖O(n)).
Clearly, the closed set F(n) is formed with a family of separated closed arcs and their condensation points. The ends of these closed arcs belong to the set
(2.24)Γ(ξ1)⋃Γ(η1),
and these condensation points belong to the set
(2.25)V=Γ(α1)⋃Γ(α2)⋃Γ(β1)⋃Γ(β2).
Obviously,(2.26)F(n)⊃F(n+1).Fact 7.
LetP(n) be the set of the ends of all separated closed arcs in F(n), and let
(2.27)B=Γ(ξ1)⋃Γ(η1).
From the Fact 3 obtained, it follows that
(2.28)P(n)⊂P(n+1)⊂B,B=⋃nP(n).
BothB and B¯ are obvious invariants under the action of Γ.Fact 8.
From the Facts3 and 4, it follows that
(2.29)B¯=V¯.Fact 9.
Clearly, by the construction of the setsO(n),
(2.30)Γ(I0)=Γ(O(0))=⋃n∈NO(n),
and for any elements a1 and a2 in Γ, if
(2.31)a1(I0)⋂a2(I0)≠∅(emptyset),
then
(2.32)a1(I0)=a2(I0).
Γ(I0) is an Γ invariant open set.Fact 10.
By the definition ofB,
(2.33)B¯⋂Γ(I0)=∅,
and from Facts 3, 4, and 8,
(2.34)V¯=B¯⊂Γ(I0)¯.Fact 11.
Let(2.35)L=V¯(=B¯).
Then from Fact 10, it is easy to see that there is no inner point in the closed set L. And from the Facts 3 and 4, it follows that
(2.36)L⊆Γ(x)¯,∀x∈S1,
and that there is no isolated point in L. Therefore, L, as a limit set of Γ, is a totally disconnected and perfect set.Fact 12.
From Facts3 and 4, it is easy to see that
(2.37)L=V¯=B¯=Γ(α1)¯=Γ(α2)¯=Γ(β1)¯=Γ(β2)¯=Γ(ξ1)¯=Γ(η1)¯.
It follows that L is a least invariant closed set of Γ.
The theorem has been proved through the above facts.


## 3. Application of the Theorem

In [13] we considered a doubly periodic Riccati equation in the complex domain: (3.1)dzdt=z2-λϱ(t),z,t∈C,
where the coefficient ϱ(t) is a Weierstrass elliptic function satisfying (3.2)[ϱ′(t)]2=4[ϱ2(t)-1]ϱ(t),ϱ(0)=0,ϱ(t+T)=ϱ(t+iT)=ϱ(t),∀t∈C,
where (3.3)T=Γ(1/2)Γ(1/4)2Γ(3/4)
is the real period of ϱ(t).Because of the doubly periodicity, (3.1) can be treated as a differential equation defined on the torus T2 [18].By the numerical results, we guessed that the solution space may have a fractal limit set whenλ<0.In [18], Guan had qualitatively proved the existence of the fractal structure for λ∈(-∞,λ0), where the critical value λ0 was evaluated approximately as −0.227.In [19], using the symmetries of (3.1), Guan et al. obtained the exact formulae of two generators E and U of the monodromy group G of (3.1), which are represented by the Möbius transformations on the extended complex plane Ĉ as follows: (3.4)E(c)=e2τc,U(c)=isinh⁡τ+ccosh⁡τcosh⁡τ-icsinh⁡τ,∀c∈Ĉ,
where the parameter τ depends on the parameter λ in (3.1) through
(3.5)sinh⁡2(τ)=cos⁡(1+4λ2π).
In addition, it is proved further that
(3.6)τ=μT,
where μ is the Floquet exponent of the related second-order linear ordinary differential equation (3.7)ü-λϱ(t)u=0.
Therefore, (3.4), (3.5), and (3.6) give the exact relation between the generators of the monodromy group, the Floquet exponent and the parameter λ in the equation. In the theory of the differential equations, this is a very rare case that these exact relations could be obtained.Now by the exact relation (3.4) and (3.5), we may see that, if (3.8)λ≤0,orλ∈[2n(2n-1),2n(2n+1)],n∈N,
then the extended imaginary axis, the left half complex plane corresponding to the negative real part, and the right half complex plane corresponding to the positive real part are all G-invariant sets. And the extended imaginary axis is just homeomorphic to S1, in which the points c1=0,c2=∞,c3=i, and c4=-i are, respectively, the fixed points of E and U. Notice that, if and only if the real parameter λ satisfies (3.9)λ<-14,
then
(3.10)cos⁡(1+4λ2π)>1,
so that, either
(3.11)e2τ>3+22,
or
(3.12)0<e2τ<3-22,
and that the commutator of E and U, H=U-1E-1UE, has exactly two different fixed points on the extended imaginary axis, (3.13)ξ1=[1-e2τ+(1-e2τ)2-4e2τ]i2e2τ,η1=[1-e2τ-(1-e2τ)2-4e2τ]i2e2τ.In this case, either the two fixed points are both located in the interval between the pointsc4=-i and c1=0, if e2τ>3+22 is hold, or they are both located in the interval between c3=i and c2=∞, if 0<e2τ<3-22 is hold. It is easy to prove further that the conditions (i)–(v) are all satisfied if λ<-1/4. Therefore, the least invariant set of G on the extended imaginary axis is totally disconnected and perfect. So, by the theorem obtained, the critical value of λ for the existence of the Cantor limit set can be exactly determined as λ0=-1/4.Now, it can be seen that, ifλ<-1/4, the doubly periodic Riccati equation (3.1) is not integrable by quadratures, and the limit set of foliation of the equation is like a Cantor book, since each point in the limit set of the monodromy group is corresponding to a piece of leaf of the foliation’s limit set [18].The related Hausdorff dimension of the least invariant set of the monodromy group is evaluated in [18] through some measure consideration.


## 4. Conclusion

The theorem obtained provides the condition for determining if a transformation group onS1 with two generators has a Cantor-like limit set. The example in Section 3 shows this theorem can be applied to the study of the complexity of the limit sets of foliations. This theorem can also be applied to the theory of discrete groups to determine if a discrete group is a Fuchsian group of the second kind [20].



---

*Source: 342759-2011-12-04.xml*