Cantor Limit Set of a Topological Transformation Group on S 1 Keying

Keying Guan1 and Zuming Chen2 1 Science College, Northern Jiao Tong University, Beijing 100044, China 2 Science College, Beijing University of Aeronautics and Astronautics, Beijing 100083, China Correspondence should be addressed to Keying Guan, keying.guan@gmail.com Received 15 June 2011; Revised 14 October 2011; Accepted 19 October 2011 Academic Editor: Rutwig Campoamor-Stursberg Copyright q 2011 K. Guan and Z. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The limit set of a topological transformation group on S1 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.


Introduction
The conception limit set of a transformation group plays an important role in both theory and application of modern mathematics.
Assume that Γ is a transformation group or semigroup formed with some continuous self-mapping on a Hausdorff space X.For any x 0 ∈ X, the set is called an orbit through x 0 under the action of Γ.For any subset A of X, let

International Journal of Mathematics and Mathematical Sciences
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.Let A be a least invariant set of Γ.For any x 0 ∈ X, if there is such a point a ∈ A that a ∈ Γ x 0 , 1.4 then For any x 0 ∈ X, a least invariant set A of Γ is called a limit set of the orbit Γ x 0 if It is easy to prove the following proposition.
Proposition 1.2.If the topology space X 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 set A 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 semigroup φ t or φ n , 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 x 0 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 corresponding monodromy group or holonomy 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 Section 2. Combining the result in 19 , the critical parameter for the existence of Cantor limit set is given exactly in Section 3.

The Theorem and Proof
Theorem 2.1.Let Γ be a transformation group on the 1-dimensional sphere, G is generated by two generators a and b, where a and b are both homeomorphic transformations onto S 1 .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 Both a and b have exactly two fixed points in S 1 , α 1 , α 2 , and β 1 , β 2 , respectively, that is, where β 1 is located in the inside of one of the two arcs of S 1 separated by α 1 and α 2 , while β 2 is located in the inside of the other one.
ii The arcs α 1 β 1 α 2 and α 1 β 2 α 2 are both invariant sets of a, that is, International Journal of Mathematics and Mathematical Sciences and the arcs β 1 α 1 β 2 and β 1 α 2 β 2 are both invariant sets of b.
iii For both i 1 and i 2,

2.4
iv The commutator of a and b, has exactly two different fixed points ξ 1 and η 1 located in the inside of the arc α 1 β 1 , that is, where ξ 1 is closer to α 1 than η 1 .In this paper, α 1 β 1 represents the open arc without two ends α 1 and β 1 , and α 1 β 1 represent, the closed arc with the two ends α 1 and β 1 , v Under the actions of a and b, the points ξ 1 and η 1 are transformed into other three arcs different to α 1 β 1 :

2.7
Proof.Let By the homeomorphic property of transformations and conditions ii and v , we can see 2.9 Let and let then the following facts and conclusions can be derived step by step.
Fact 1.The closed set F 0 is formed with following four separated closed arcs:

2.13
Fact 3. From i , ii , and iii , it follows that

2.15
Fact 4. Clearly, the commutator h of a and b is also a homeomorphic transformation.From the condition iv , it follows that, either

2.18
Fact 5. Let Z 0 represent the set of all nonzero integers, and let then the family of transformations {a m } m∈Z 0 changes F β into a family of separated closed arcs in F α , which are condensed at α 1 and α 2 , and the family of transformation {b m } m∈Z 0 changes F α into a family of separated closed arcs in F β with β 1 and β 2 as their condensation points.

and let
2.21 then it follows that

2.22
For an integer n greater than 1, we may inductively let

2.23
International Journal of Mathematics and Mathematical Sciences 7 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 and these condensation points belong to the set

2.26
Fact 7. Let P n be the set of the ends of all separated closed arcs in F n , and let

2.27
From the Fact 3 obtained, it follows that

2.35
Then from Fact 10, it is easy to see that there is no inner point in the closed set L. And from the Fact 3 and 4, it follows that 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 Facts 3 and 4, it is easy to see that

2.37
It follows that L is a least invariant closed set of Γ.
The theorem has been proved through the above facts.

Application of the Theorem
In 13 we considered a doubly periodic Riccati equation in the complex domain: where the coefficient t is a Weierstrass elliptic function satisfying

3.2
International Journal of Mathematics and Mathematical Sciences 9 where is the real period of t .Because of the doubly periodicity, 3.1 can be treated as a differential equation defined on the torus T 2 18 .
By the numerical results, we guessed that the solution space may have a fractal limit set when λ < 0.
In 18 , Guan had qualitatively proved the existence of the fractal structure for λ ∈ −∞, λ 0 , where the critical value λ 0 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 M öbius transformations on the extended complex plane C as follows: where the parameter τ depends on the parameter λ in 3. 1 − e 2τ − 1 − e 2τ 2 − 4e 2τ i 2e 2τ .

3.13
In this case, either the two fixed points are both located in the interval between the points c 4 −i and c 1 0, if e 2τ > 3 2 √ 2 is hold, or they are both located in the interval between c 3 i and c 2 ∞, if 0 < e 2τ < 3 − 2 √ 2 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 λ 0 −1/4.Now, it can be seen that, if λ < −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.

Conclusion
The theorem obtained provides the condition for determining if a transformation group on S 1 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 .
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 International Journal of Mathematics and Mathematical Sciences G-invariant sets.And the extended imaginary axis is just homeomorphic to S 1 , in which the points c 1 0, c 2 ∞, c 3 i, and c 4 −i are, respectively, the fixed points of E and U. Notice that, if and only if the real parameter λ satisfies