On Topological Classification of Regular Denjoy Type Homeomorphisms

Abstract

We consider regular Denjoy type homeomorphisms of the two-dimensional torus which are the most natural generalization of Denjoy homeomorphisms of the circle. In particular, they arise as Poincaré maps induced on global cross sections by leaves of one-dimensional orientable unstable foliations of some partially hyperbolic diffeomorphisms of closed three-dimensional manifolds. The nonwandering set of each regular Denjoy type homeomorphism is a Sierpiński set, and each such homeomorphism is, by definition, semiconjugate to the minimal translation on the two-dimensional torus. For regular Denjoy type homeomorphisms, we introduce a complete invariant of topological conjugacy characterized by the minimal translation, which is semiconjugate to the given regular Denjoy type homeomorphism, with a distinguished at most countable set of orbits.

Let X be a topological space and $f:X \to X$ be a homeomorphism. A nonempty subset M of X is called a minimal set of f if M is closed and invariant under f (i.e., $f(M) = M)$ and does not have nonempty closed invariant subsets other than M. If the entire space X is a minimal set of f, then f is called minimal. Well-known examples of minimal homeomorphisms are minimal rotations of the circle and minimal translations on the two-dimensional torus.¹

Homeomorphisms of the circle with no periodic points that are topologically nonconjugate to a minimal rotation were first considered in [13]. Later, such diffeomorphisms (homeomorphisms) of the circle were called Denjoy diffeomorphisms (homeomorphisms). The nonwandering set of a Denjoy homeomorphism is minimal and homeomorphic to a Cantor set. Moreover, each such homeomorphism is semiconjugate to a minimal rotation and the complete preimage of each point of the circle under a semiconjugate map is either a point or homeomorphic to a closed interval. A topological classification of Denjoy homeomorphisms was obtained in [9].

Maps of the 2-torus that have properties characteristic of Denjoy homeomorphisms of the circle were considered in [10–12]. According to [11], we introduce following definition.

Definition 1. A homeomorphism $f:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ is called a Denjoy type homeomorphism if it satisfies the following conditions:

1. f is semiconjugate to a minimal translation $g:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ under a continuous map $p:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ homotopic to identity (i.e., $p \circ f = g \circ p$).

2. The set $B = \{ x \in {{\mathbb{T}}^{2}}:{{p}^{{ - 1}}}(x)$ contains more than one point}² is nonempty and countable.

The set B will be called the characteristic set of f. Note that, if a point x belongs to B, then all points of its orbit under the map g also belong to B.

It should be noted that the direct product of two Denjoy homeomorphisms of the circle is not a Denjoy type homeomorphism of the 2-torus.

Let $f:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ be a Denjoy type homeomorphism. Then, by virtue of [11], the complete preimage of each point $x \in {{\mathbb{T}}^{2}}$ under a semiconjugate map p is connected and the nonwandering set of f is minimal. However, in contrast to Denjoy homeomorphisms of the circle, nonwandering sets of Denjoy type homeomorphisms may not be homeomorphic (in induced topologies). In this paper, we consider the subclass of Denjoy type homeomorphisms of the 2-torus (see Definition 2 below) with homeomorphic nonwandering sets. Such homeomorphisms are the most natural generalization of Denjoy homeomorphisms of the circle and admit a complete topological classification, which is obtained in Theorem 1 below.

Definition 2. A Denjoy type homeomorphism $f:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ is called regular if the complete preimage of each point of its characteristic set under a semiconjugate map p is a closed embedded disk³ and the diameters of these disks form a sequence converging to zero.

A partially hyperbolic diffeomorphism h of the 3-torus ${{\mathbb{T}}^{3}}$ that has a two-dimensional attractor was constructed in [10]. Specifically, it was obtained from an algebraic Anosov automorphism via a bifurcation of birth of an invariant curve (similar constructions were considered in [3, 5, 7]). According to [10], a one-dimensional orientable unstable foliation of h has a global cross section (2-torus) and its leaves induce on it a Poincaré map that is a regular Denjoy type homeomorphism.

Below, we describe the topological properties of nonwandering sets of regular Denjoy type homeomorphisms that underlie the proof of the present results.

Definition 3. A Sierpiński set on the 2-torus ${{\mathbb{T}}^{2}}$ is a set $S = {{\mathbb{T}}^{2}}\backslash \bigcup\limits_{k \in \mathbb{Z}} \,{\text{int}}{\kern 1pt} {{D}_{k}}$, where ${{\{ {{D}_{k}}\} }_{{k \in \mathbb{Z}}}}$ is the family of sets with the following properties:

(1) for each $k \in \mathbb{Z}$, the set D_k is a closed embedded disk;

(2) $\bigcup\limits_{k \in \mathbb{Z}} \,{\text{int}}{\kern 1pt} {{D}_{k}}$ is dense in ${{\mathbb{T}}^{2}}$;

(3) ${{D}_{k}} \cap {{D}_{{k'}}} = \emptyset$ for $k \ne k{\kern 1pt} '$;

(4) ${\text{diam}}({{D}_{k}})$, $k \in \mathbb{Z}$, forms a sequence converging to zero.

Let $S = {{\mathbb{T}}^{2}}{{\backslash }}\bigcup\limits_{k \in \mathbb{Z}} \,{\text{int}}{\kern 1pt} {{D}_{k}}$ be a Sierpiński set on the 2-torus ${{\mathbb{T}}^{2}}$. For every $k \in \mathbb{Z}$, let L_k denote the boundary of the set D_k. Since D_k is a closed embedded disk, L_k is a simple closed curve. Let $I = S{{\backslash }}\bigcup\limits_{k \in \mathbb{Z}} {{L}_{k}}$. Each point $x \in I$ is called an interior point of S, and I is called the set of interior points of S.

Let Q be a standard Sierpiński carpet on the square $V = [0;1] \times [0;1]$ (for the construction, see, e.g., [2, pp. 275–276]). A set C on ${{\mathbb{T}}^{2}}$ is defined as $\pi {{|}_{V}}(Q)$, where $\pi {{|}_{V}}$ is the restriction of the natural projection $\pi :{{\mathbb{R}}^{2}} \to {{\mathbb{T}}^{2}}$ to the square V. Then C is a Sierpiński set. The set of interior points of C is denoted by ${{I}_{C}}$. By virtue of [4, 14], for any Sierpiński set S, there exists a homeomorphism $\theta :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ such that $\theta (S) = C$ and $\theta (I) = {{I}_{C}}$.

The following result is given without proof.

Lemma 1. The nonwandering set of a regular Denjoy type homeomorphism of the 2-torus is a Sierpiński set.

According to [1], a map $\varphi :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ is called linear if it can be represented as the composition of an algebraic automorphism and a translation on the 2‑torus.

Let f₁ and f₂ be regular Denjoy type homeomorphisms of the 2-torus such that f_j $(j \in \{ 1,2\} )$ is semiconjugate to a minimal translation ${{g}_{j}}:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ under the map ${{p}_{j}}:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$. Let ${{B}_{j}}$ be the characteristic set of the homeomorphism f_j.

Theorem 1. Let f₁ and f₂ be regular Denjoy type homeomorphisms of the 2-torus. Then f₁ and f₂ are topologically conjugate if and only if there exists a linear map $\varphi :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ such that $\varphi \circ {{g}_{1}} = {{g}_{2}} \circ \varphi$ and $\varphi ({{B}_{1}}) = {{B}_{2}}$.

The key element in the proof of Theorem 1 is the proof of sufficiency, i.e., the construction of a homeomorphism $\psi :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ that is a modification of $\varphi$ under which  f₁ and f₂ are conjugate. Schematically, this construction consists of the following five steps.

1. Let ${{S}_{1}}$ and ${{S}_{2}}$ denote the nonwandering sets of  f₁ and f₂, respectively, and ${{I}_{1}}$ and ${{I}_{2}}$ denote the sets of interior points of ${{S}_{1}}$ and ${{S}_{2}}$, respectively. It can be directly verified that the restriction of the map p_j, $j \in \{ 1,2\}$, to I_j is a homeomorphism of I_j to the image ${{p}_{j}}({{I}_{j}})$.

2. According to [4, 14], there exist homeomorphisms ${{\theta }_{1}}:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ and ${{\theta }_{2}}:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ such that ${{\theta }_{1}}({{S}_{1}}) = C$, ${{\theta }_{1}}({{I}_{1}}) = {{I}_{C}}$ and ${{\theta }_{2}}({{S}_{2}}) = C$, ${{\theta }_{2}}({{I}_{2}}) = {{I}_{C}}$. Then the map $h:{{I}_{C}} \to {{I}_{C}}$ defined as h(x) = ${{\theta }_{2}}(p_{2}^{{ - 1}}(\varphi ({{p}_{1}}(\theta _{1}^{{ - 1}}(x)))))$, where $x \in {{I}_{C}}$, is a homeomorphism.

3. There exists a homeomorphism $\zeta :C \to C$ such that $\zeta (x) = h(x)$ for all $x \in {{I}_{C}}$. This follows from the uniform continuity of the map h, which is proved below.

The map $\kappa :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ is defined as κ(x) = ${{\theta }_{1}}(p_{1}^{{ - 1}}({{\varphi }^{{ - 1}}}({{p}_{2}}(\theta _{2}^{{ - 1}}(x)))))$, where $x \in {{\mathbb{T}}^{2}}$ and $p_{1}^{{ - 1}}({{\varphi }^{{ - 1}}}({{p}_{2}}(\theta _{2}^{{ - 1}}(x))))$ is the complete preimage of the set ${{\varphi }^{{ - 1}}}({{p}_{2}}(\theta _{2}^{{ - 1}}(x)))$. Note that $\kappa (x) = {{h}^{{ - 1}}}(x)$ for all $x \in {{I}_{C}}$.

For each positive integer n, let $Q_{1}^{n},...,Q_{{{{8}^{n}}}}^{n}$ denote equal squares obtained at the nth step of constructing the Sierpiński carpet Q (see [2, pp. 275–276]), and let $K_{1}^{n},...,K_{{{{8}^{n}}}}^{n}$ denote the images of $Q_{1}^{n},...,Q_{{{{8}^{n}}}}^{n}$ under the map $\pi {{|}_{V}}$, respectively. Since $Q \subset \bigcup\limits_{i = 1}^{{{8}^{n}}} Q_{i}^{n}$ (for each $n \in \mathbb{N}$), it is true that $C \subset \bigcup\limits_{i = 1}^{{{8}^{n}}} K_{i}^{n}$ (for each $n \in \mathbb{N}$).

For a fixed $\varepsilon > 0,$ we choose $m \in \mathbb{N}$ $(m \geqslant 2)$ such that ${\text{diam}}(K_{i}^{m}) < \frac{\varepsilon }{2}$. For each i, the set $\tilde {K}_{i}^{m}$ is defined as $\tilde {K}_{i}^{m} = \kappa (K_{i}^{m})$.

Let A and $B$ be sets on the 2-torus ${{\mathbb{T}}^{2}}$. The distance between A and $B$ is defined as dist(A, B) = ${\text{inf}}\{ \rho (x,y)|x \in A,y \in B\}$, where $\rho$ denotes the distance between points on ${{\mathbb{T}}^{2}}$ induced by the Riemannian metric.

For each set $\tilde {K}_{i}^{m},$ the quantity d_i is defined as d_i = $\mathop {\min }\limits_j ({\text{dist}}(\tilde {K}_{i}^{m},\tilde {K}_{j}^{m}))$, where $\tilde {K}_{j}^{m}$ is a set that does not share points with $\tilde {K}_{i}^{m}$. Choose $\delta > 0$ such that δ < $\mathop {\min }\limits_{i \in {{{\{ 1,...,8}}^{m}}\} } {{d}_{i}}$. Any two points ${{x}_{1}},{{x}_{2}} \in {{I}_{C}}$ such that $\rho ({{x}_{1}},{{x}_{2}})$ < δ are contained in either a single set $\tilde {K}_{i}^{m}$ or two different sets $\tilde {K}_{i}^{m}$ and $\tilde {K}_{j}^{m}$ sharing at least one point. Then $h({{x}_{1}})$ and $h({{x}_{2}})$ lie either in a single set $K_{i}^{m}$ or two different sets $K_{i}^{m}$ and $K_{j}^{m}$ sharing at least one point. From this result and the relation ${\text{diam}}(K_{i}^{m})\, < \,\frac{\varepsilon }{2},$ it follows that $\rho (h({{x}_{1}}),h({{x}_{2}})) < \varepsilon$. Thus, the map h is uniformly continuous on the set ${{I}_{C}}$.

4. The map $\chi :{{S}_{1}} \to {{S}_{2}}$ is defined as χ(x) = $\theta _{2}^{{ - 1}}(\zeta ({{\theta }_{1}}(x)))$, where $x \in {{S}_{1}}$. Then ${{p}_{2}} \circ {{f}_{2}}{{{\text{|}}}_{{{{I}_{2}}}}}$ = ${{g}_{2}} \circ {{p}_{2}}{{{\text{|}}}_{{{{I}_{2}}}}}$ = $\varphi \circ {{g}_{1}} \circ {{\varphi }^{{ - 1}}} \circ {{p}_{2}}{{{\text{|}}}_{{{{I}_{2}}}}}$ = $\varphi \circ {{g}_{1}} \circ {{p}_{1}} \circ {{\chi }^{{ - 1}}}{{{\text{|}}}_{{{{I}_{2}}}}}$ = $\varphi \circ {{p}_{1}}$ $\circ$ ${{f}_{1}} \circ {{\chi }^{{ - 1}}}{{{\text{|}}}_{{{{I}_{2}}}}}$ = ${{p}_{2}} \circ \chi \circ {{f}_{1}} \circ {{\chi }^{{ - 1}}}{{{\text{|}}}_{{{{I}_{2}}}}}$. Thus, $\chi \circ {{f}_{1}}{{{\text{|}}}_{{{{I}_{1}}}}} = {{f}_{2}} \circ \chi {{{\text{|}}}_{{{{I}_{1}}}}}$. Since the maps $\chi ,{{f}_{1}},{{f}_{2}}$ are continuous and I₁ is dense in S₁, it follows that $\chi \circ {{f}_{1}}{{{\text{|}}}_{{{{S}_{1}}}}} = {{f}_{2}} \circ \chi {{{\text{|}}}_{{{{S}_{1}}}}}$.

5. Lemma 1 and the equality $\varphi ({{B}_{1}}) = {{B}_{2}}$ imply that the homeomorphism χ extends to a homeomorphism $\psi :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ such that $\psi \circ {{f}_{1}} = {{f}_{2}} \circ \psi$.

Thus, the proof of the sufficiency of the conditions in Theorem 1 is complete.

Theorem 1 implies the following result.

Corollary 1. Let f₁ and f₂ be regular Denjoy type homeomorphisms of the 2-torus such that the characteristic set of each of them consists of a single orbit. Then  f₁ and f₂ are topologically conjugate if and only if there exists an algebraic automorphism $\eta :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ such that $\eta \circ {{g}_{1}} = {{g}_{2}} \circ \eta$.

The necessity follows immediately from Theorem 1. Let us prove the sufficiency.

Let ${{O}_{1}}$ and ${{O}_{2}}$ be orbits that are the characteristic sets of  f₁ and f₂, respectively. Since $\eta \circ {{g}_{1}} = {{g}_{2}} \circ \eta$, the automorphism $\eta$ maps the orbits of ${{g}_{1}}$ to the orbits of ${{g}_{2}}$. Then we can choose a translation $\gamma :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ such that the map $\varphi = \gamma \circ \eta$ takes the orbit ${{O}_{1}}$ to ${{O}_{2}}$. By construction, $\varphi$ is a linear map. Moreover, the translations ${{g}_{1}}$ and ${{g}_{2}}$ are conjugate under $\varphi$. Then the sufficiency follows from Theorem 1.

Following [8], for any minimal translation g and any set B consisting of $n$ orbits g $(n \geqslant 1)$, there exists a regular Denjoy type homeomorphism that is semiconjugate to g and its characteristic set coincides with B. From Theorem 1 and the results of [8], it follows that there exists a standard representative in each class of topological conjugacy of regular Denjoy type homeomorphisms with characteristic sets consisting of finitely many orbits. We do not know whether an example of a regular Denjoy type homeomorphism with a characteristic set consisting of a countable number of orbits has been constructed.⁴

Theorem 2. For any minimal translation $g\,:\,{{\mathbb{T}}^{2}}\, \to \,{{\mathbb{T}}^{2}}$ and any positive integer $n \geqslant 2,$ there exists a continual set of pairwise topologically nonconjugate regular Denjoy type homeomorphisms of the 2-torus, each being semiconjugate to g and having a characteristic set consisting of $n$ orbits.

The idea of the proof of Theorem 2 can be described as follows.

Let $g:{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ be a minimal translation and $n \geqslant 2$ be a positive integer. By Theorem 1 and the results of [8], it suffices to show that there exists a continuum of sets ${{B}_{\nu }}$ having the following properties:

(1) Each set ${{B}_{\nu }}$ is the union of n orbits of g.

(2) For any two sets ${{B}_{{{{\nu }_{1}}}}}$ and ${{B}_{{{{\nu }_{2}}}}}$, there does not exist a linear map $\varphi :{{\mathbb{T}}^{2}} \to {{\mathbb{T}}^{2}}$ such that $\varphi ({{B}_{{{{\nu }_{1}}}}}) = {{B}_{{{{\nu }_{2}}}}}$.

Let ${{O}_{1}},...,{{O}_{{n - 1}}}$ be n – 1 arbitrary different orbits of g. Define the set ${{B}_{\nu }} = \bigcup\limits_{i = 1}^{n - 1} {{O}_{i}} \cup {{O}_{\nu }}$, where ${{O}_{\nu }}$ is an orbit of g other than ${{O}_{1}},...,{{O}_{{n - 1}}}$. By direct verification, we can show that it is possible to choose a continuum of different orbits ${{O}_{\nu }}$ and, hence, a continuum of different sets ${{B}_{\nu }}$ such that the sets ${{B}_{\nu }}$ satisfy the required conditions.