Cantor Limit Set of a Topological Transformation Group on S¹

Abstract

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.

1. 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 𝑥₀∈𝑋, the set Γ𝑥₀=𝛾𝑥₀∣𝛾∈Γ (1.1) is called an orbit through 𝑥₀ under the action of Γ. For any subset 𝐴 of 𝑋, let Γ(𝐴)=_𝑥∈𝐴Γ(𝑥). (1.2) A subset 𝐴 of 𝑋 is called a Γ-invariant set if Γ(𝐴)=𝐴. (1.3) 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 𝑥₀∈𝑋, if there is such a point 𝑎∈𝐴 that 𝑎∈Γ𝑥₀, (1.4) then 𝐴⊂Γ𝑥₀. (1.5)

For any 𝑥₀∈𝑋, a least invariant set 𝐴 of Γ is called a limit set of the orbit Γ(𝑥₀) if 𝐴⊂Γ𝑥₀. (1.6) 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 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 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.

2. The Theorem and Proof

Theorem 2.1 .. Let Γ be a transformation group on the 1-dimensional sphere, 𝐒¹=𝑅𝑍. (2.1)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)Both 𝐚 and 𝐛 have exactly two fixed points in 𝐒¹,𝛼₁,𝛼₂, and 𝛽₁,𝛽₂, respectively, that is, 𝐚𝛼_𝑖=𝛼_𝑖,𝐛𝛽_𝑖=𝛽_𝑖,𝑖=1,2, (2.2) 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 arcs 𝛼₁𝛽₁𝛼₂ and 𝛼₁𝛽₂𝛼₂ are both invariant sets of 𝐚, that is, 𝐚:𝛼₁𝛽₁𝛼₂⟼𝛼₁𝛽₁𝛼₂,𝐚:𝛼₁𝛽₂𝛼₂⟼𝛼₁𝛽₂𝛼₂, (2.3) and the arcs 𝛽₁𝛼₁𝛽₂ and 𝛽₁𝛼₂𝛽₂ are both invariant sets of 𝐛.(𝑖𝑖𝑖)For both 𝑖=1 and 𝑖=2, lim_𝑛→∞𝐚^𝑛𝛽_𝑖=𝛼₂,lim_𝑛→∞𝐚^−𝑛𝛽_𝑖=𝛼₁,lim_𝑛→∞𝐛^𝑛𝛼_𝑖=𝛽₂,lim_𝑛→∞𝐛^−𝑛𝛼_𝑖=𝛽₁. (2.4)(𝑖𝑣)The commutator of 𝐚 and 𝐛, 𝐡=𝐛⁻¹𝐚⁻¹𝐛𝐚(∈Γ), (2.5) has exactly two different fixed points 𝜉₁ and 𝜂₁ located in the inside of the arc 𝛼₁𝛽₁, that is, 𝜉₁,𝜂₁∈𝛼₁𝛽₁. (2.6) 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 of 𝐚 and 𝐛, the points 𝜉₁ and 𝜂₁ are transformed into other three arcs different to 𝛼₁𝛽₁: 𝜉₂=𝐚𝜉₁,𝜂₂=𝐚𝜂₁,𝜉₂,𝜂₂∈𝛽₁𝛼₂,𝜉₃=𝐛𝜉₂,𝜂₃=𝐛𝜂₂,𝜉₃,𝜂₃∈𝛽₂𝛼₂,𝜉₄=𝐛𝜉₁,𝜂₄=𝐛𝜂₁,𝜉₄,𝜂₄∈𝛼₁𝛽₂. (2.7)

Proof. Let 𝐼₀=(𝜉₁𝜂₁). (2.8) By the homeomorphic property of transformations and conditions (ii) and (v), we can see 𝜉₂𝜂₂=𝐚𝐼₀,𝜉₃𝜂₃=𝐛𝐚𝐼₀,𝜉₄𝜂₄=𝐛𝐼₀=𝐚⁻¹𝐛𝐚𝐼₀. (2.9) Let 𝑂(0)=𝐼₀𝐚𝐼₀𝐛𝐚𝐼₀𝐛𝐼₀, (2.10) and let 𝐹(0)=𝐒¹⧵𝑂(0); (2.11) 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: 𝐹_𝛼1=𝜂₄𝛼₁𝜉₁,𝐹_𝛼2=𝜂₂𝛼₂𝜉₃,𝐹_𝛽1=𝜂₁𝛽₁𝜉₂,𝐹_𝛽2=𝜂₃𝛽₂𝜉₄. (2.12)Fact 2. Obviously, 𝐚𝐹_𝛼1=𝐹_𝛽2𝐹_𝛽1𝐹_𝛼1𝐼₀𝑏𝐼₀,𝐚⁻¹𝐹_𝛼2=𝐹_𝛽2𝐹_𝛽1𝐹_𝛼2𝐚𝐼₀𝐛𝐚𝐼₀,𝐛𝐹_𝛽1=𝐹_𝛼2𝐹_𝛼1𝐹_𝛽1𝐼₀𝐚𝐼₀,𝐛⁻¹𝐹_𝛽2=𝐹_𝛼2𝐹_𝛼1𝐹_𝛽2𝐛𝐼₀𝐛𝐚𝐼₀. (2.13)Fact 3. From (i), (ii), and (iii), it follows that lim_𝑛→∞𝐚^𝑛(𝑥)=𝛼₂,lim_𝑛→∞𝐚^−𝑛(𝑥)=𝛼₁,∀𝑥∈𝐒¹⧵𝛼₁,𝛼₂, (2.14) and that lim_𝑛→∞𝐛^𝑛(𝑥)=𝛽₂,lim_𝑛→∞𝐛^−𝑛(𝑥)=𝛽₁,∀𝑥∈𝐒¹⧵𝛽₁,𝛽₂. (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 lim_{𝑛→−∞}𝐡^𝑛(𝑥)=𝜉₁,lim_𝑛→+∞𝐡^𝑛(𝑥)=𝜂₁ (2.16) or lim_{𝑛→−∞}𝐡^𝑛(𝑥)=𝜂₁,lim_𝑛→+∞𝐡^𝑛(𝑥)=𝜉₁ (2.17) if 𝑥∈𝐒¹⧵𝜉₁,𝜂₁. (2.18)Fact 5. Let 𝑍₀ represent the set of all nonzero integers, and let 𝐹_𝛼=𝐹_𝛼1𝐹_𝛼2,𝐹_𝛽=𝐹_𝛽1𝐹_𝛽2; (2.19) then the family of transformations {𝐚^𝑚}_𝑚∈𝑍0 changes 𝐹_𝛽 into a family of separated closed arcs in 𝐹_𝛼, which are condensed at 𝛼₁ and 𝛼₂, and the family of transformation {𝐛^𝑚}_𝑚∈𝑍0 changes 𝐹_𝛼 into a family of separated closed arcs in 𝐹_𝛽 with 𝛽₁ and 𝛽₂ as their condensation points.Fact 6. Let 𝐹_𝛼(1)=_𝑚∈𝑍0𝐚^𝑚𝐹_𝛽⊂𝐹_𝛼,𝐹_𝛽(1)=_𝑚∈𝑍0𝐛^𝑚𝐹_𝛼⊂𝐹_𝛽, (2.20) and let 𝑂(1)=_𝑚∈𝑍𝐚^𝑚(𝑂(0))𝐛^𝑚(𝑂(0,𝐹(1)=))𝐹_𝛼(1)𝐹_𝛽(1); (2.21) then it follows that 𝐹(1)=𝐒¹⧵𝑂(1). (2.22)
For an integer 𝑛 greater than 1, we may inductively let 𝐹_𝛼(𝑛)=_𝑚∈𝑍0𝐚^𝑚𝐹_𝛽(𝑛−1)⊂𝐹_𝛼(),𝐹𝑛−1_𝛽(𝑛)=_𝑚∈𝑍0𝐚^𝑚𝐹_𝛼(𝑛−1)⊂𝐹_𝛽(),𝑂(𝑛)=𝑛−1_𝑚∈𝑍𝐚^𝑚(𝑂(𝑛−1))𝐛^𝑚(𝑂((),𝐹(𝑛)=𝑛−1))⊃𝑂𝑛−1𝐹_𝛼(𝑛)𝐹_𝛽(𝑛)=𝐒¹⧵𝑂(𝑛). (2.23) 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 Γ𝜉₁Γ𝜂₁, (2.24) and these condensation points belong to the set 𝑉=Γ𝛼₁Γ𝛼₂Γ𝛽₁Γ𝛽₂. (2.25)
Obviously, 𝐹(𝑛)⊃𝐹(𝑛+1). (2.26)Fact 7. Let 𝑃(𝑛) be the set of the ends of all separated closed arcs in 𝐹(𝑛), and let 𝐵=Γ𝜉₁Γ𝜂₁. (2.27) From the Fact 3 obtained, it follows that 𝑃(𝑛)⊂𝑃(𝑛+1)⊂𝐵,𝐵=_𝑛𝑃(𝑛). (2.28)
Both 𝐵 and 𝐵 are obvious invariants under the action of Γ.Fact 8. From the Facts 3 and 4, it follows that 𝐵=𝑉. (2.29)Fact 9. Clearly, by the construction of the sets 𝑂(𝑛), Γ𝐼₀=Γ(𝑂(0))=_𝑛∈𝑁𝑂(𝑛), (2.30) and for any elements 𝐚₁ and 𝐚₂ in Γ, if 𝐚₁𝐼₀𝐚₂𝐼₀≠∅emptyset, (2.31) then 𝐚₁𝐼₀=𝐚₂𝐼₀. (2.32)
Γ(𝐼₀) is an Γ invariant open set.Fact 10. By the definition of 𝐵, 𝐵Γ𝐼₀=∅, (2.33) and from Facts 3, 4, and 8, 𝑉=𝐵⊂Γ𝐼₀. (2.34)Fact 11. Let 𝐿=𝑉=𝐵. (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 Facts 3 and 4, it follows that 𝐿⊆Γ(𝑥),∀𝑥∈𝑆¹, (2.36) 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 𝐿=𝑉=𝐵=Γ𝛼₁=Γ𝛼₂=Γ𝛽₁=Γ𝛽₂=Γ𝜉₁=Γ𝜂₁. (2.37) 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: 𝑑𝑧𝑑𝑡=𝑧²−𝜆𝜚(𝑡),𝑧,𝑡∈𝐶, (3.1) where the coefficient 𝜚(𝑡) is a Weierstrass elliptic function satisfying 𝜚^′(𝑡)²=4𝜚²(𝑡)𝜚(𝑡)(0)𝜚()()(𝑡)−1,𝜚=0,𝑡+𝑇=𝜚𝑡+𝑖𝑇=𝜚,∀𝑡∈𝐶, (3.2) where Γ()Γ()𝑇=1/21/42Γ(3/4) (3.3) 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 when 𝜆<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 𝐶 as follows: 𝐸(𝑐)=𝑒^2𝜏𝑈(𝑐)=𝑐,𝑖sinh𝜏+𝑐cosh𝜏,cosh𝜏−𝑖𝑐sinh𝜏∀𝑐∈𝐶, (3.4) where the parameter 𝜏 depends on the parameter 𝜆 in (3.1) through sinh²(𝜏)√=cos1+4𝜆2𝜋. (3.5) In addition, it is proved further that 𝜏=𝜇𝑇, (3.6) where 𝜇 is the Floquet exponent of the related second-order linear ordinary differential equation 𝑢̈−𝜆𝜚(𝑡)𝑢=0. (3.7) 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 𝜆≤0,or𝜆∈[2𝑛(2𝑛−1),2𝑛(2𝑛+1)],𝑛∈𝑁, (3.8) 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 1𝜆<−4, (3.9) then √cos1+4𝜆2𝜋>1, (3.10) so that, either 𝑒^2𝜏>3+2√2, (3.11) or 0<𝑒^2𝜏<3−2√2, (3.12) and that the commutator of 𝐸 and 𝑈, 𝐻=𝑈⁻¹𝐸⁻¹𝑈𝐸, has exactly two different fixed points on the extended imaginary axis, 𝜉₁=1−𝑒^2𝜏+1−𝑒^2𝜏2−4𝑒^2𝜏𝑖2𝑒^2𝜏,𝜂₁=1−𝑒^2𝜏−1−𝑒^2𝜏2−4𝑒^2𝜏𝑖2𝑒^2𝜏. (3.13)

In this case, either the two fixed points are both located in the interval between the points 𝑐₄=−𝑖 and 𝑐₁=0, if 𝑒^2𝜏>3+2√2 is hold, or they are both located in the interval between 𝑐₃=𝑖 and 𝑐₂=∞, 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 𝜆₀=−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.

4. Conclusion

The theorem obtained provides the condition for determining if a transformation group on 𝐒¹ 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].

Acknowledgment

This project is supported by the National Natural Science Foundation of China.