Classification of special Anosov endomorphisms of nil-manifolds

TL;DR

This paper classifies special endomorphisms of nil-manifolds, showing that such maps are topologically conjugate to hyperbolic nil-endomorphisms when they are special TA-maps.

Contribution

It provides a classification result for special endomorphisms of nil-manifolds, linking them to hyperbolic nil-endomorphisms via topological conjugacy.

Findings

Special endomorphisms are conjugate to hyperbolic nil-endomorphisms.

The classification applies to special TA-maps on nil-manifolds.

Homotopic maps share conjugacy properties.

Abstract

In this paper we give a classification of special endomorphisms of nil-manifolds: Let $f:N/\Gamma \rightarrow N/\Gamma$ be a covering map of a nil-manifold and denote by $A:N/\Gamma \rightarrow N/\Gamma$ the nil-endomorphism which is homotopic to $f$. If $f$ is a special $TA$-map, then $A$ is a hyperbolic nil-endomorphism and $f$ is topologically conjugate to $A$.

Equations

$$X_f=\{\tilde{x}=(x_i):x_i\in X\text{ and }f(x_i)=x_{i+1},i\in \mathbb{Z}\},$$

$${\sigma }_f((x_i))=(f(x_i)).$$

$$f^{-1}(V_y)=\bigcup _i{U}_i(i\ne i^{\prime }\Rightarrow U_i\cap U_i^{\prime }=\varnothing )$$

$$T_{x_i}M=E_{x_i}^s\oplus E_{x_i}^u,i\in \mathbb{Z}$$

$$\parallel D_{x_i}{f}^n(v)\parallel \le C{\lambda }^n\parallel v\parallel \text{ if }v\in E_{x_i}^s,$$

$$\parallel D_{x_i}{f}^n(v)\parallel \ge C^{-1}{\lambda }^{-n}\parallel v\parallel \text{ if }v\in E_{x_i}^u.$$

$$W^s(x)$$

$$W^u(\tilde{x})$$

$$x\sim y\iff y=g\cdot x\text{ for some }g\in G$$

$$\forall (n,\gamma )\in Aff(N),\forall x\in N:(n,\gamma ).x=n\gamma (x).$$

$$\forall n_1,n_2\in N,\forall {\psi }_1,{\psi }_2\in Aut(N):n_1{\psi }_1{n}_2{\psi }_2=n_1{\psi }_1(n_2){\psi }_1{\psi }_2.$$

$$A:N/\Gamma \to N/\Gamma ,\Gamma \cdot n\mapsto \Gamma \cdot \overline{\mathcal{A}}(n).$$

$$A:N/\Gamma \to N/\Gamma ,\Gamma \cdot n\mapsto \Gamma \cdot \overline{A}(n).$$

$$\overline{\mathcal{A}}:N⋊F\to N⋊F:x\mapsto \varphi x{\varphi }^{-1}\text{ (conjugation in }Aff(N))$$

$${\pi }_1(X;x_0)=\Omega (X;x_0)/\sim .$$

$$w_{♯}:\Omega (X;x_1)\to \Omega (X;x_0),\text{ by }{w}_{♯}(u)=(w\cdot u)\cdot \overline{w},$$

$$w_{\ast }:{\pi }_1(X;x_1)\to {\pi }_1(X;x_0),\text{ by }{w}_{\ast }([u])=[w_{♯}(u)],$$

$$w(t)=F(x_0,t)t\in [0,1],$$

$$f_{♯}:\Omega (X;x_0)\to \Omega (Y;Y_0),\text{ by }{f}_{♯}(u)=fu,$$

$$f_{\ast }:{\pi }_1(X;x_0)\to {\pi }_1(Y;y_0),\text{ by }{f}_{\ast }([u])=[f_{♯}(u)]=[fu],$$

$$\Gamma \cong {\pi }_1(N/\Gamma )\cong G(\pi ).$$

$$f([e])=f\circ \pi (e)=\pi \circ \overline{f}(e)=\pi (\overline{f}(e))=\pi (e)=[e]=A([e]).$$

$$||D_e{\overline{A}}^n(v)||\le c{\lambda }^n||v||(v\in E_e^s),$$

$$||D_e{\overline{A}}^{-n}(v)||\le c{\lambda }^n||v||(v\in E_e^u),$$

$${\overline{L}}^s(x)=\{y\in N:D({\overline{A}}^i(x),{\overline{A}}^i(y))\to 0(i\to \infty )\},$$

$${\overline{L}}^u(x)=\{y\in N:D({\overline{A}}^i(x),{\overline{A}}^i(y))\to 0(i\to -\infty )\}.$$

$${\overline{W}}^s(x)\subset U_{{\delta }_K}({\overline{L}}^s(\overline{h}(x)))\text{and}{\overline{W}}^u(x;\text{e})\subset U_{{\delta }_K}({\overline{L}}^u(\overline{h}(x))).$$

$${\overline{W}}^s(x)$$

$${\overline{W}}^u(x,\text{e})$$

$$D(x^{-1},e)$$

$$D$$

$$=D(x,e)$$

$$D({\overline{f}}^i(y),{\overline{f}}^i(x))<D({\overline{f}}^i(y),{\overline{f}}^i(e))+D({\overline{f}}^i(e),{\overline{f}}^i(x))\to 0\text{ as }i\to -\infty .$$

$$D({\overline{f}}^i(y),{\overline{f}}^i(e))<D({\overline{f}}^i(y),{\overline{f}}^i(x))+D({\overline{f}}^i(x),{\overline{f}}^i(e))\to 0\text{ as }i\to -\infty .$$

Full text

1. Introduction

Finding a universal model for Anosov diffeomorphisms has been an important problem in dynamical systems. In this general context, Franks and Manning proved that every Anosov diffeomorphism of an infra-nil-manifold is topologically conjugate to a hyperbolic infra-nil-automorphism [7, 8, 12, 13] (According to Dekimpe’s work [5], some of their results are incorrect). Based on this result, Aoki and Hiraide has been studied the dynamics of covering maps of a torus [2]. The importance of infra-nil-manifolds comes from the following Conjecture 1.1 and Theorem 1.2 :

The first non-toral example of an Anosov diffeomorphism was constructed by S. Smale in [16]. He conjectured that, up to topologically conjugacy, the construction in Smale’s example gives every possible Anosov diffeomorphism on a closed manifold.

Conjecture 1.1**.**

Every Anosov diffeomorphism of a closed manifold is topologically conjugate to a hyperbolic affine infra-nil-automorphism.

Theorem 1.2** (Gromov [9]).**

Every expanding map on a closed manifold is topologically conjugate to an expanding affine infra-nil-endomorphism.

The conjecture has been open for many years (see [6] page 48). An interesting problem is to consider the conjecture for endomorphisms of a closed manifold. Our main theorem is a partial answer to the conjecture.

In this paper we give a classification of special endomorphisms of nil-manifolds. Infact, Aoki and Hiraide [2] in 1994 proposed two problems:

Problem 1.3**.**

Is every special Anosov differentiable map of a torus topologically conjugate to a hyperbolic toral endomorphism?

Problem 1.4**.**

Is every special topological Anosov covering map of an arbitrary closed topological manifold topologically conjugate to a hyperbolic infra-nil- endomorphism of an infra-nil-manifold ?

Aoki and Hiraide answered problem 1.3 partially as follows:

Theorem 1.5** ([2] Theorem 6.8.1).**

Let $f:\mathbb{T}^{n}\rightarrow\mathbb{T}^{n}$ be a $TA$-covering map of an $n$-torus and denote by $A:\mathbb{T}^{n}\rightarrow\mathbb{T}^{n}$ the toral endomorphism homotopic to $f$. Then $A$ is hyperbolic. Furthermore the inverse limit system of $(\mathbb{T}^{n},f)$ is topologically conjugate to the inverse limit system of $(\mathbb{T}^{n},A)$.

Theorem 1.6** ([2] Theorem 6.8.2).**

Let $f$ and $A$ be as Theorem 1.5. Suppose $f$ is special, then the following statements hold:

1. (1)

if $f$ is a $TA$-homeomorphism, then $A$ is a hyperbolic toral automorphism and $f$ is topologically conjugate to $A$, 2. (2)

if $f$ is a topological expanding map, then $A$ is an expanding toral endomorphism and $f$ is topologically conjugate to $A$, 3. (3)

if $f$ is a strongly special $TA$-map, then $A$ is a hyperbolic toral endomorphism and $f$ is topologically conjugate to $A$.

In [17], Sumi has altered the condition ” strongly special ” (part (3) of Theorem 1.6) to just ” special ” as follows:

Theorem 1.7** ([17]).**

Let $f$ and $A$ be as Theorem 1.5. If $f$ is a special $TA$-map, then $A$ is a hyperbolic toral endomorphism and $f$ is topologically conjugate to $A$.

In [18], Sumi generalized (incorrectly) Theorem 1.5 and parts (1) and (2) of Theorem 1.6 for infra-nil-manifolds as follows:

Theorem 1.8** ([18] Theorem 1).**

Let $f:N/\Gamma\rightarrow N/\Gamma$ be a covering map of an infra-nil-manifold and denote as $A:N/\Gamma\rightarrow N/\Gamma$ the infra-nil-endomorphism homotopic to $f$. If $f$ is a TA-map, then $A$ is hyperbolic and the inverse limit system of $(N/\Gamma,f)$ is topologically conjugate to the inverse limit system of $(N/\Gamma,A)$ .

Theorem 1.9** ([18] Theorem 2).**

Let $f$ and $A$ be as in Theorem 1.8. Then the following statements hold:

1. (1)

if $f$ is a TA-homeomorphism, then $A$ is a hyperbolic infra-nil-automorphism and $f$ is topologically conjugate to $A$, 2. (2)

if $f$ is a topological expanding map, then $A$ is an expanding infra-nil-endomorphism and $f$ is topologically conjugate to $A$.

Dekimpe [5], expressed that there might exist (interesting) diffeomorphisms and self-covering maps of an infra-nil-manifold which are not even homotopic to an infra-nil-endomorphism. Dekimpe [5] in §4, gave an expanding map not topologically conjugate to an infra-nil-endomorphism. And in §5, he gave an Anosov diffeomorphism not topologically conjugate to an infra-nil-automorphism. According to [5], Theorem 1.8 and Theorem 1.9 are true for nil-manifolds. Of course, if in Sumi’s works, the map $f$ has a desired homotopic infra-nil-endomorphism, then the theorems hold.

Since, nil-manifolds are included in infra-nil-manifolds, we consider [18] for nil-manifolds.

In the paper, by using Theorem 1.7, we partially answer problem 1.4 of Aoki and Hiraide as follows:

Theorem 1.10** (Main Theorem).**

Let $f:N/\Gamma\rightarrow N/\Gamma$ be a covering map of a nil-manifold and denote as $A:N/\Gamma\rightarrow N/\Gamma$ the nil-endomorphism homotopic to $f$ (according to [5], such a unique homotopy exists for nil-manifolds). If $f$ is a special $TA$-map, then $A$ is a hyperbolic nil-endomorphism and $f$ is topologically conjugate to $A$.

Corollary 1.11**.**

If $f:N/\Gamma\rightarrow N/\Gamma$ is a special Anosov endomorphism of a nil-manifold then it is conjugate to a hyperbolic nil-endomorphism.

2. Preliminaries

Let $X$ and $Y$ be compact metric spaces and let $f:X\rightarrow X$ and $g:Y\rightarrow Y$ be continuous surjections. Then $f$ is said to be topologically conjugate to $g$ if there exists a homeomorphism $\varphi:Y\rightarrow X$ such that $f\circ\varphi=\varphi\circ g$.

Let $X$ be a compact metric space with metric $d$ . For $f:X\rightarrow X$ a continuous surjection, we let

[TABLE]

The map $\sigma_{f}:X_{f}\rightarrow X_{f}$ is called the shift map determined by $f$. We call $(X_{f},\sigma_{f})$ the inverse limit of $(X,f)$. A homeomorphism $f:X\rightarrow X$ is called expansive if there is a constant $e>0$ (called an expansive constant) such that if $x$ and $y$ are any two distinct points of $X$ then $d(f^{i}(x),f^{i}(y))>e$ for some integer $i$. A continuous surjection $f:X\rightarrow X$ is called $c$-expansive if there is a constant $e>0$ such that for $\tilde{x},\tilde{y}\in X_{f}$ if $d(x_{i},y_{i})\leq e$ for all $i\in\mathbb{Z}$ then $\tilde{x}=\tilde{y}$. In particular, if there is a constant $e>0$ such that for $x,y\in X$ if $d(f^{i}(x),f^{i}(y))\leq e$ for all $i\in\mathbb{N}$ then $x=y$, we say that $f$ is positively expansive. A sequence of points $\{x_{i}:a<i<b\}$ of $X$ is called a $\delta$-pseudo orbit of $f$ if $d(f(x_{i}),x_{i+1})<\delta$ for $i\in(a,b-1)$. Given $\epsilon>0$ a $\delta$-pseudo orbit of $\{x_{i}\}$ is called to be $\epsilon$-traced by a point $x\in X$ if $d(f^{i}(x),x_{i})<\epsilon$ for every $i\in(a,b-1)$ . Here the symbols $a$ and $b$ are taken as $-\infty\leq a<b\leq\infty$ if $f$ is bijective and as $-1\leq a<b\leq\infty$ if $f$ is not bijective. $f$ has the pseudo orbit tracing property (abbrev. POTP) if for every $\epsilon>0$ there is $\delta>0$ such that every $\delta$-pseudo orbit of $f$ can be $\epsilon$-traced by some point of $X$.

We say that a homeomorphism $f:X\rightarrow X$ is a topological Anosov map (abbrev. $TA$-map) if $f$ is expansive and has POTP. Analogously, We say that a continuous surjection $f:X\rightarrow X$ is a topological Anosov map if $f$ is $c$-expansive and has POTP, and say that $f$ is a topological expanding map if $f$ is positively expansive and open. We can check that every topological expanding map is a $TA$-map (see [2] Remark 2.3.10).

Let $X$ and $Y$ be metric spaces. A continuous surjection $f:X\rightarrow Y$ is called a covering map if for $y\in Y$ there exists an open neighborhood $V_{y}$ of $y$ in $Y$ such that

[TABLE]

where each of $U_{i}$ is open in $X$ and $f_{|U_{i}}:U_{i}\rightarrow V_{y}$ is a homeomorphism. A covering map $f:X\rightarrow Y$ is especially called a self-covering map if $X=Y$. We say that a continuous surjection $f:X\rightarrow Y$ is a local homeomorphism if for $x\in X$ there is an open neighborhood $U_{x}$, of $x$ in $X$ such that $f(U_{x})$ is open in $Y$ and $f_{|U_{x}}:U_{x}\rightarrow f(U_{x})$ is a homeomorphism. It is clear that every covering map is a local homeomorphism. Conversely, if $X$ is compact, then a local homeomorphism $f:X\rightarrow Y$ is a covering map (see [2] Theorem 2.1.1).

Let $\pi:Y\rightarrow X$ be a covering map. A homeomorphism $\alpha:Y\rightarrow Y$ is called a covering transformation for $\pi$ if $\pi\circ\alpha=\pi$ holds. We denote as $G(\pi)$ the set of all covering transformations for $\pi$. It is easy to see that $G(\pi)$ is a group, which is called the covering transformation group for $\pi$.

Let $M$ be a closed smooth manifold and let $C^{1}(M,M)$ be the set of all $C^{1}$ maps of $M$ endowed with the $C^{1}$ topology. A map $f\in C^{1}(M,M)$ is called an Anosov endomorphism if $f$ is a $C^{1}$ regular map and if there exist $C>0$ and $0<\lambda<1$ such that for every $\tilde{x}=(x_{i})\in M_{f}=\{\tilde{x}=(x_{i}):x_{i}\in M\text{ and }f(x_{i})=x_{i+1},\;i\in\mathbb{Z}\}$ there is a splitting

[TABLE]

(we show this by $T_{\tilde{x}}M=\bigcup_{i}(E_{x_{i}}^{s}\oplus E_{x_{i}}^{u})$) so that for all $i\in\mathbb{Z}$:

1. (1)

$D_{x_{i}}f(E_{x_{i}}^{\sigma})=E_{x_{i+1}}^{\sigma}$ where $\sigma=s,u$, 2. (2)

for all $n\geq 0$

[TABLE]

If, in particular, $T_{\tilde{x}}M=\bigcup_{i}E_{x_{i}}^{u}$ for all $\tilde{x}=(x_{i})\in M_{f}$ , then $f$ is said to be expanding differentiable map, and if an Anosov endomorphism $f$ is injective then $f$ is called an Anosov diffeomorphism. We can check that every Anosov endomorphism is a TA-map, and that every expanding differentiable map is a topological expanding map (see [2] Theorem 1.2.1).

A map $f\in C^{1}(M,M)$ is said to be $C^{1}$-structurally stable if there is an open neighborhood $\mathcal{N}(f)$ of $f$ in $C^{1}(M,M)$ such that $g\in\mathcal{N}(f)$ implies that $f$ and $g$ are topologically conjugate. Anosov [1] proved that every Anosov diffeomorphism is $C^{1}$-structurally stable, and Shub [15] showed the same result for expanding differentiable maps. However, Anosov endomorphisms which are not diffeomorphisms nor expanding do not be $C^{1}$-structurally stable ([11],[14]).

A map $f\in C^{1}(M,M)$ is said to be $C^{1}$-inverse limit stable if there is an open neighborhood $\mathcal{N}(f)$ of $f$ in $C^{1}(M,M)$ such that $g\in\mathcal{N}(f)$ implies that the inverse limit $(M_{f},\sigma_{f})$ of $(M,f)$ and the inverse limit $(M_{g},\sigma_{g})$ of $(M,g)$ are topologically conjugate. Mané and Pugh [11] proved that every Anosov endomorphism is $C^{1}$-inverse limit stable.

We define special TA-maps as follows. Let $f:X\rightarrow X$ be a continuous surjection of a compact metric space. Define the stable and unstable sets

[TABLE]

for $x\in X$ and $\tilde{x}\in X_{f}$. A TA-map $f:X\rightarrow X$ is special if $f$ satisfies the property that $W^{u}(\tilde{x})=W^{u}(\tilde{y})$ for every $\tilde{x},\tilde{y}\in X_{f}$ with $x_{0}=y_{0}$. Every hyperbolic nil-endomorphism is a special TA-covering map (See [18] Remark 3.13). By this and Theorem 1.10 We have the following corollary:

Corollary 2.1**.**

A TA-covering map of a nil-manifold is special if and only if it is conjugate to a hyperbolic nil-endomorphism.

A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. Let $N$ be a Lie group. A vector field $X$ on $N$ is said to be invariant under left translations if for each $g,h\in N,\;(dl_{g})_{h}(X_{h})=X_{gh}$, where $(dl_{g})_{h}:T_{h}N\rightarrow T_{gh}N$ and $l_{g}:N\rightarrow N;\;x\mapsto gx$. A Lie algebra $g$ is a vector space over some field $F$ together with a binary operation $[\cdot,\cdot]:g\times g\rightarrow g$ called the Lie bracket, that satisfies:

1. (1)

Bilinearity: $[ax+by,z]=a[x,z]+b[y,z],\;[z,ax+by]=a[z,x]+b[z,y]\;\forall x,y,z\in g$, 2. (2)

Alternativity: $[x,x]=0\;\forall x\in g$, 3. (3)

The Jacobi Identity: $[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0\;\forall x,y,z\in g$

Let $Lie(N)$ be the set of all left-translation-invariant vector fields on $N$. It is a real vector space. Moreover, it is closed under Lie bracket. Thus $Lie(N)$ is a Lie subalgebra of the Lie algebra of all vector fields on $N$ and is called the Lie algebra of $N$. A nilpotent Lie group is a Lie group which is connected and whose Lie algebra is a nilpotent Lie algebra. That is, its Lie algebra’s central series eventually vanishes.

A group $G$ is a torsion group if every element in $G$ is of finite order. $G$ is called torsion free if no element other than identity is of finite order. A discrete subgroup of a topological group $G$ is a subgroup $H$ such that there is an open cover of $H$ in which every open subset contains exactly one element of $H$. In other words, the subspace topology of $H$ in $G$ is the discrete topology. A uniform subgroup $H$ of $G$ is a closed subgroup such that the quotient space $G/H$ is compact.

We bring here the definitions of nil-manifolds and infra-nil-manifolds from Karel Dekimpe in [4] and [5].

Let $N$ be a Lie group and $Aut(N)$ be the set of all automorphisms of $N$. Assume that $\overline{A}\in Aut(N)$ is an automorphism of $N$, such that there exists a discrete and cocompact subgroup $\Gamma$ of $N$, with $\overline{A}(\Gamma)\subseteq\Gamma$. Then the space of left cosets $N/\Gamma$ is a closed manifold, and $\overline{A}$ induces an endomorphism $A:N/\Gamma\rightarrow N/\Gamma,\;g\Gamma\mapsto\overline{A}(g)\Gamma$. If we want this endomorphism to be Anosov, $\overline{A}$ must be hyperbolic (i.e. has no eigenvalue with modulus 1). It is known that this can happen only when $N$ is nilpotent. So we restrict ourselves to that case, where the resulting manifold $N/\Gamma$ is said to be a nil-manifold. Such an endomorphism $A$ induced by an automorphism $\overline{A}$ is called a nil-endomorphism and is said to be a hyperbolic nil-automorphism, when $\overline{A}$ is hyperbolic. If in the above definition, $\overline{A}(\Gamma)=\Gamma$, the induced map is called a nil-automorphism.

All tori, $\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$ are examples of nil-manifolds.

Let $X$ be a topological space and let $G$ be a group. We say that $G$ acts (continuously) on $X$ if to $(g,x)\in G\times X$ there corresponds a point $g\cdot x$ in $X$ and the following conditions are satisfied:

1. (1)

$e\cdot x=x$ for $x\in X$ where $e$ is the identity, 2. (2)

$g\cdot(g^{\prime}\cdot x)=gg^{\prime}\cdot x$ for $x\in X$ and $g,g^{\prime}\in G$, 3. (3)

for each $g\in G$ a map $x\mapsto g\cdot x$ is a homeomorphism of $X$.

When $G$ acts on $X$, for $x,y\in X$ letting

[TABLE]

an equivalence relation $\sim$ in $X$ is defined. Then the identifying space $X/\sim$, denoted as $X/G$, is called the orbit space by $G$ of $X$. It follows that for $x\in X$, $[x]=\{g\cdot x:g\in G\}$ is the equivalence class.

An action of $G$ on $X$ is said to be properly discontinuous if for each $x\in X$ there exists a neighborhood $U(x)$ of $x$ such that $U(x)\cap gU(x)=\emptyset$ for all $g\in G$ with $g\neq e_{G}$. Here $gU(x)=\{g\cdot y:y\in U(x)\}$.

Now we give an extended definition of nil-manifolds. Let $N$ be a connected and simply connected nilpotent Lie group and $Aut(N)$ be the group of continuous automorphisms of $N$. Then $Aff(N)=N\rtimes Aut(N)$ acts on $N$ in the following way:

[TABLE]

So an element of $Aff(N)$ consists of a translational part $n\in N$ and a linear part $\gamma\in Aut(N)$ (as a set $Aff(N)$ is just $N\times Aut(N)$) and $Aff(N)$ acts on $N$ by first applying the linear part and then multiplying on the left by the translational part). In this way, $Aff(N)$ can also be seen as a subgroup of $Diff(N)$.

Now, let $C$ be a compact subgroup of $Aut(N)$ and consider any torsion free discrete subgroup $\Gamma$ of $N\rtimes C$, such that the orbit space $N/\Gamma$ is compact. Note that $\Gamma$ acts on $N$ as being also a subgroup of $Aff(N)$. The action of $\Gamma$ on $N$ will be free and properly discontinuous, so $N/\Gamma$ is a manifold, which is called an infra-nil-manifold.

Klein bottle is an example of infra-nil-manifolds.

In what follows, we will identify $N$ with the subgroup $N\times\{id\}$ of $N\rtimes Aut(N)=Aff(N)$, $F$ with the subgroup $\{id\}\times F$ and $Aut(N)$ with the subgroup $\{id\}\times Aut(N)$.

It follows from Theorem 1 of L. Auslander in [3], that $\Gamma\cap N$ is a uniform lattice of $N$ and that $\Gamma/(\Gamma\cap N)$ is a finite group. This shows that the fundamental group of an infra-nil-manifold $N/\Gamma$ is virtually nilpotent (i.e. has a nilpotent normal subgroup of finite index). In fact $\Gamma\cap N$ is a maximal nilpotent subgroup of $\Gamma$ and it is the only normal subgroup of $\Gamma$ with this property. (This also follows from [3]).

If we denote by $p:N\rtimes C\rightarrow C$ the natural projection on the second factor, then $p(\Gamma)=\Gamma\cap N$ is a uniform lattice of $N$ and that $\Gamma/(\Gamma\cap N)$. Let $F$ denote this finite group $p(\Gamma)$, then we will refer to $F$ as being the holonomy group of $\Gamma$ (or of the infra-nil-manifold $N/\Gamma$). It follows that $\Gamma\subseteq N\rtimes F$. In case $F=\{id\}$, so $\Gamma\subseteq N$, the manifold $N/\Gamma$ is a nil-manifold. Hence, any infra-nil-manifold $N/\Gamma$ is finitely covered by a nil-manifold $N/(\Gamma\cap N)$. This also explains the prefix ”infra”.

Fix an infra-nil-manifold $N/\Gamma$, so $N$ is a connected and simply connected nilpotent Lie group and $\Gamma$ is a torsion free, uniform discrete subgroup of $N\rtimes F$, where $F$ is a finite subgroup of $Aut(N)$. We will assume that $F$ is the holonomy group of $\Gamma$ (so for any $\mu\in F$, there exists an $n\in N$ such that $(n,\mu)\in\Gamma)$.

We can say that an element of $\Gamma$ is of the form $n\mu$ for some $n\in N$ and some $\mu\in F$. Also, any element of $Aff(N)$ can uniquely be written as a product $n\psi$, where $n\in N$ and $\psi\in Aut(N)$. The product in $Aff(N)$ is then given as

[TABLE]

Now we can define infra-nil-endomorphisms as follows:

Let $N$ be a connected, simply connected nilpotent Lie group and $F\subseteq Aut(N)$ a finite group. Assume that $\Gamma$ is a torsion free, discrete and uniform subgroup of $N\rtimes F$. Let $\overline{\mathcal{A}}:N\rtimes F\rightarrow N\rtimes F$ be an automorphism, such that $\overline{\mathcal{A}}(F)=F$ and $\overline{\mathcal{A}}(\Gamma)\subseteq\Gamma$, then, the map

[TABLE]

is the infra-nil-endomorphism induced by $\overline{\mathcal{A}}$. In case $\overline{\mathcal{A}}(\Gamma)=\Gamma$, we call $A$ an infra-nil-automorphism.

In the definition above, $\Gamma\cdot n$ denotes the orbit of n under the action of $\Gamma$. The computation above shows that $A$ is well defined. Note that infra-nil-automorphisms are diffeomorphisms, while in general an infra-nil-endomorphism is a self-covering map.

The following theorem shows that the only maps of an infra-nil-manifold, that lift to an automorphism of the corresponding nilpotent Lie group are exactly the infra-nil-endomorphisms defined above.

Theorem 2.2** ([5] Theorem 3.4).**

Let $N$ be a connected and simply connected nilpotent Lie group, $F\subseteq Aut(N)$ a finite group and $\Gamma$ a torsion free discrete and uniform subgroup of $N\rtimes F$ and assume that the holonomy group of $\Gamma$ is $F$. If $\overline{A}:N\rightarrow N$ is an automorphism for which the map

[TABLE]

is well defined (meaning that $\Gamma\cdot\overline{A}(n)=\Gamma\cdot\overline{A}(\gamma\cdot n)$ for all $\gamma\in\Gamma$), then

[TABLE]

is an automorphism of $N\rtimes F$, with $\overline{\mathcal{A}}(F)=F$ and $\overline{\mathcal{A}}(\Gamma)\subseteq\Gamma$. Hence, $A$ is an infra-nil-endomorphism.

Let $X$ be a topological space. We write $\Omega(X;x_{0})$ the family of all closed paths from $x_{0}$ to $x_{0}$. Let $\Omega(X;x_{0})/\sim$ be the identifying space with respect to the equivalence relation $\sim$ by homotopty. We write this set

[TABLE]

The group $\pi_{1}(X;x_{0})$ is called the fundamental group at a base point $x_{0}$ of $X$. If, in particular, $\pi_{1}(X;x_{0})$ is a group consisting of the identity, then $X$ is said to be simply connected with respect to a base point $x_{0}$.

Let $x_{0}$ and $x_{1}$ be points in $X$. If there exists a path $w$ joining $x_{0}$ and $x_{1}$, then we can define a map

[TABLE]

where $u\in\Omega(X;x_{1})$, $(w\cdot u)$ is the concatenation of $w$ and $u$ and $\overline{w}$ is $w$ in reverse direction. For $u,v\in\Omega(X;x_{1})$ suppose $u\sim v$. Then $w_{\sharp}(u)\sim w_{\sharp}(v)$ and thus $w_{\sharp}$ induces a map

[TABLE]

this map is an isomorphism (see [2] Lemma 6.1.4).

Remark 2.3*.*

If $X$ is a path connected space then we can remove the base point and write $\pi_{1}(X;x_{0})=\pi_{1}(X)$.

Let $f,g:X\rightarrow Y$ be homotopic and $F$ a homotopy from $f$ to $g$ ($f\sim g\;(F)$). Then for $x_{0}\in X$ we can define a path $w\in\Omega(Y;f(x_{0}),g(x_{0}))$ by

[TABLE]

and the relation between homomorphisms $f_{*}:\pi_{1}(X;x_{0})\rightarrow\pi_{1}(Y;f(x_{0}))$ and $g_{*}:\pi_{1}(X;x_{0})\rightarrow\pi_{1}(Y;g(x_{0}))$ is: $g_{*}=\overline{w}_{*}\circ f_{*}$ (see [2] Lemma 6.1.9).

Let $X$ and $Y$ be topological spaces and $f:X\rightarrow Y$ a continuous map. Take $x_{0}\in X$ and let $y_{0}=f(x_{0})$. It is clear that $fu=f\circ u\in\Omega(X,y_{0})$ for $u\in\Omega(X;x_{0})$. Thus we can find a map

[TABLE]

where $u\in\Omega(X;x_{0})$. If $u\sim v\;(F)$ for $u,v\in\Omega(X;x_{0})$, then we have $fu\sim fv\;(f\circ F)$, from which the following map will be induced:

[TABLE]

It is easy to check that $f_{*}$ is a homomorphism. We say that $f_{*}$ is a homomorphism induced from a continuous map $f:X\rightarrow Y$.

Lemma 2.4** ([2] Remark 6.7.9).**

Let $f,g:N/\Gamma\rightarrow N/\Gamma$ be continuous maps of a nil-manifold and let $f(x_{0})=g(x_{0})$ for some $x_{0}\in N/\Gamma$. Then $f$ and $g$ are homotopic if and only if $f_{*}=g_{*}:\pi_{1}(N/\Gamma,x_{0})\rightarrow\pi_{1}(N/\Gamma,f(x_{0}))$.

Theorem 2.5** ([2] Theorem 6.3.4).**

If $\pi:Y\rightarrow X$ is the universal covering, then for each $b\in Y$

1. (1)

the map $\alpha\mapsto\alpha(b)$ is a bijection from $G(\pi)$ onto $\pi^{-1}(\pi(b))$, 2. (2)

the map $\psi_{b}:G(\pi)\rightarrow\pi_{1}(X,\pi(b))$ by $\alpha\mapsto[\pi\circ u_{\alpha(b)}]$ is an isomorphism where $u_{\alpha(b)}$ is a path from $b$ to $\alpha(b)$.

Furthermore, the action of $G(\pi)$ on $Y$ is properly discontinuous and $Y/G(\pi)$ is homeomorphic to $X$ .

Theorem 2.6** ([2] Theorem 6.3.7).**

Let $G$ be a group and $X$ a topological space. Suppose that $G$ acts on $X$ and the action is properly discontinuous. Then

1. (1)

the natural projection $\pi:X\rightarrow X/G$ is a covering map, 2. (2)

if $X$ is simply connected, then the fundamental group $\pi_{1}(X/G)$ is isomorphic to $G$.

Corollary 2.7**.**

Let $N/\Gamma$ be an infra-nil-manifold and $\pi:N\rightarrow N/\Gamma$ be the natural projection. Then

[TABLE]

Proof.

Since $\Gamma$ acts on $N$ properly discontinuous, the natural projection $\pi:N\rightarrow N/\Gamma$ is a covering map. Since $N$ is simply connected, by Theorem 2.6 we have $\Gamma\cong\pi_{1}(N/\Gamma)$.

On the other hand, since $N$ is simply connected and $\Gamma$ acts on $N$ properly discontinuous the natural projection $\pi:N\rightarrow N/\Gamma$ is the universal covering map. So by Theorem 2.5 we have $\Gamma\cong G(\pi)$. ∎

From now on we only consider $N/\Gamma$ as a nil-manifold.

Lemma 2.8**.**

Let $f:N/\Gamma\rightarrow N/\Gamma$ be a continuous map of a nil-manifold, and $A:N/\Gamma\rightarrow N/\Gamma$ be the unique nil-endomorphism homotopic to $f$, then $\overline{f}_{*}=\overline{A}_{*}:\Gamma\rightarrow\Gamma$.

Proof.

By corollary 2.7, $\overline{f}_{*}$ and $\overline{A}_{*}$ are two maps on $\Gamma$. For $[e]=\{x\in N:\gamma(x)=\gamma.x=e\text{ for some }\gamma\in\Gamma\}$, we have

[TABLE]

So according to lemma 2.4, $\overline{f}_{*}=\overline{A}_{*}$. ∎

Lemma 2.9** ([18] Lemma 1.3).**

Let $f:N/\Gamma\rightarrow N/\Gamma$ be a self-covering map of a nil-manifold and $A:N/\Gamma\rightarrow N/\Gamma$ denote the nil-endomorphism homotopic to $f$. If $f$ is a TA-covering map, then $A$ is hyperbolic.

Lemma 2.10** ([18] Lemma 1.5).**

Let $f:N/\Gamma\rightarrow N/\Gamma$ be a self-covering map and let $\overline{f}$ : $N\rightarrow N$ be a lift of $f$ by the natural projection $\pi:N\rightarrow N/\Gamma$. If $f$ is a TA-covering map then $\overline{f}$ has exactly one fixed point.

For continuous maps $f$ and $g$ of $N$ we define $D(f,g)=\sup\{D(f(x),g(x)):x\in N\}$ where $D$ denotes a left invariant, $\Gamma$-invariant Riemannian distance for $N$ . Notice that $D(f,g)$ is not necessary finite.

Suppose that $f$ : $N/\Gamma\rightarrow N/\Gamma$ is a TA-covering map. Let $A:N/\Gamma\rightarrow N/\Gamma$ be the nil-endomorphism homotopic to $f$, and let $\overline{A}$ : $N\rightarrow N$ be the automorphism which is a lift of $A$ by the natural projection $\pi$. Since $D_{e}\overline{A}$ is hyperbolic by Lemma 2.9, the Lie algebra $Lie(N)$ of $N$ splits into the direct sum $Lie(N)=E_{e}^{s}\oplus E_{e}^{u}$ of subspaces $E_{e}^{s}$ and $E_{e}^{u}$ such that $D_{e}\overline{A}(E_{e}^{s})=E_{e}^{s}$, $D_{e}\overline{A}(E_{e}^{u})=E_{e}^{u}$ and there are $c>1,0<\lambda<1$ so that for all $n\geq 0$

[TABLE]

where $||\cdot||$ is the Riemannian metric. Let $\overline{L}^{\sigma}(e)=\exp(E_{e}^{\sigma})\;(\sigma=s,u)$ and let $\overline{L}^{\sigma}(x)=x\cdot\overline{L}^{\sigma}(e)(\sigma=s,u)$ for $x\in N$ . Since left translations are isometries under the metric $D$, it follows that for all $x\in N$

[TABLE]

Lemma 2.11** ([10] Lemma 3.2).**

For $x,y\in N$, $L^{s}(X)\cap{L^{u}(y)}$ consists of exactly one point.

For $x,y\in N$ denote as $\beta(x,y)$ the point in $L^{s}(X)\cap{L^{u}(y)}$ .

Lemma 2.12** ([10] Lemma 3.2).**

*(1) For $L>0$ and $\epsilon>0$ there exists $J>0$ such that for $x,y\in N$ if $D(\overline{A}^{i}(x),\overline{A}^{i}(y))\leq L$ for all $i$ with $|i|\leq J$ , then $D(x,y)\leq\epsilon$.

(2) For given $L>0$, if $D(\overline{A}^{i}(x),\overline{A}^{i}(y))\leq L$ for all $i\in\mathbb{Z}$ , then $x=y\;(x,y\in N)$.*

Lemma 2.13** ([18] Lemma 2.3).**

Under the assumptions and notations as above, there is a unique map $\overline{h}$ : $N\rightarrow N$ such that

1. (1)

$\overline{A}\circ\overline{h}=\overline{h}\circ\overline{f}$, 2. (2)

$D(\overline{h},id_{N})$* is finite,*

where $id_{N}$ : $N\rightarrow N$ is the identity map of $N$. Furthermore $\overline{h}$ is surjective and uniformly continuous under $D$ .

In addition, if $f$ is not an expanding map then $\overline{h}$ is a homeomorphism i.e. $\overline{h}$ is $D$-biuniformly continuous. (See [2] Proposition 8.4.2)

Lemma 2.14** ([18] Lemma 2.4).**

For the semiconjugacy $\overline{h}$ of lemma 2.13, we have the following properties:

1. (1)

There exists $K>0$ such that $D(\overline{h}(x\gamma),\overline{h}(x)\gamma)<K$ for $x\in\mathbb{N}$ and $\gamma\in\Gamma$. 2. (2)

For any $\lambda>0$ , there exists $L\in\mathbb{N}$ such that $D(\overline{h}(x\gamma),\overline{h}(x)\gamma)<\lambda$ for $x\in N$ and $\gamma\in\overline{A}_{*}^{L}(\Gamma)$ . 3. (3)

For $x\in N$ and $\gamma\in\bigcap_{i=0}^{\infty}\overline{A}_{*}^{i}(\Gamma)$ , we have $\overline{h}(x\gamma)=\overline{h}(x)\gamma$ . 4. (4)

For $x\in N$ and $\gamma\in\Gamma$, we have $\overline{h}(x\gamma)\in\overline{L}^{s}(\overline{h}(x)\gamma)$.

Remark 2.15*.*

By part (2) of theorem 2.13, there is a $\delta_{K}>0$ such that $D(\overline{h}(x),x)<\delta_{K}$ for $x\in N$, we have (see [2] page 270 (8.5))

[TABLE]

By lemma 2.10 if $f:N/\Gamma\rightarrow N/\Gamma$ is a $TA$-map and $\overline{f}:N\rightarrow N$ a lift of it, then there exists a unique fixed point say $b$ such that $\overline{f}(b)=b$. For simplisity we can suppose that $b=e$. Indeed, we can choose a homeomorphism $\varphi$ of $N$ such that $\varphi(\pi(b))=e$. Then $\varphi\circ f\circ\varphi^{-1}$ is a $TA$-covering map such that $\varphi\circ f\circ\varphi^{-1}(e)=e$.

Let $x\in N$, we define the stable set and unstable sets of $x$ for $f$ and $A$ as follow (for more details see [2]):

[TABLE]

Where $\textbf{e}=(\ldots,e,e,e,\ldots)$.

Remark 2.16*.*

By lemma 2.13, since $\overline{h}$ is $D$-uniformly continuous then $\overline{h}(\overline{W}^{s}(x))=\overline{L}^{s}(\overline{h}(x))$ and $\overline{h}(\overline{W}^{u}(x;\textbf{e}))=\overline{L}^{u}(\overline{h}(x))$.

Lemma 2.17**.**

The following statements hold:

1. (1)

$\overline{W}^{s}(x)\gamma=\overline{W}^{s}(x\gamma)$* for $\gamma\in\Gamma$ and $x\in N$,* 2. (2)

$\overline{W}^{u}(x;\textbf{e})\gamma=\overline{W}^{u}(x\gamma;\textbf{e})$* for $\gamma\in\Gamma$ and $x\in N$,*

Proof.

It is an easy corollary of lemma 6.6.11 of [2]. According to corollary 2.7, in the mentioned lemma put $\Gamma$ instead of $G(\pi)$ and $N$ instead of $\overline{X}$. ∎

Lemma 2.18**.**

The following statements hold:

1. (1)

$\overline{L}^{s}(x)\gamma=\overline{L}^{s}(x\gamma)$* for $\gamma\in\Gamma$ and $x\in N$,* 2. (2)

$\overline{L}^{u}(x)\gamma=\overline{L}^{u}(x\gamma)$* for $\gamma\in\Gamma$ and $x\in N$,*

Proof.

Proof is the same as in lemma 2.17. ∎

Lemma 2.19** ([18] Lemma 5.4).**

Let $N/\Gamma$ be a nil-manifold. If $f:N/\Gamma\rightarrow N/\Gamma$ is a $TA$-covering map, then the nonwandering set $\Omega(f)$ coincides with the entire space $N/\Gamma$.

Lemma 2.20** ([2] Lemma 8.6.2).**

For $\epsilon>0$ there is $\delta>0$ such that if $D(x,y)<\delta,\;x,y\in N$ then $\overline{W}^{s}(x)\subset U_{\epsilon}(\overline{W}^{s}(y))$ and $\overline{W}^{u}(x;\textbf{e})\subset U_{\epsilon}(\overline{W}^{u}(y;\textbf{e}))$. Where for a set $S,\;U_{\epsilon}(S)=\{y\in N:D(y,S)<\epsilon\}$.

3. Proof of Main Theorem

In this section we suppose that $f:N/\Gamma\rightarrow N/\Gamma$ is a special $TA$-covering map of a nil-manifold which is not injective or expanding, and $A:N/\Gamma\rightarrow N/\Gamma$ is the unique nil-endomorphism homotopic to $f$.

Sketch of proof. By lemma 2.13, there is a unique semiconjugacy $\overline{h}:N\rightarrow N$ between $\overline{f}$ and $\overline{A}$, such that by proposition 3.2.(3), $\overline{h}(v\gamma)=\overline{h}(v)\gamma$, for each $\gamma\in\overline{W}^{u}(e;\textbf{e})\cap\Gamma$ and $v\in\overline{W}^{u}(e;\textbf{e})$. Through proposition 3.3 to proposition 3.13 we show that for all $\gamma\in\Gamma$ and $x\in N$, $\overline{h}(x\gamma)=\overline{h}(x)\gamma$. Based on this result, $\overline{h}$ induces a homeomorphism $h:N/\Gamma\rightarrow N/\Gamma$ which is the conjugacy between $f$ and $A$.

To prove the main theorem we need some consequential lemmas and propositions.

Lemma 3.1**.**

The following statements hold:

1. (1)

Let $D$ be the metric of $N$ as above. for each $x\in N$, $D(x^{-1},e)=D(x,e)$. 2. (2)

If $x\in\overline{W}^{u}(e;\textbf{e})$, then $\overline{W}^{u}(x;\textbf{e})=\overline{W}^{u}(e;\textbf{e})$. 3. (3)

If $x\in\overline{L}^{u}(e)$, then $\overline{L}^{u}(x)=\overline{L}^{u}(e)$.

Proof.

(1)

[TABLE]

(2) Since $x\in\overline{W}^{u}(e;\textbf{e})$, we have $D(\overline{f}^{i}(x),\overline{f}^{i}(e))\rightarrow 0$ as $i\rightarrow-\infty$. Let $y\in\overline{W}^{u}(e;\textbf{e})$, then $D(\overline{f}^{i}(y),\overline{f}^{i}(e))\rightarrow 0$ as $i\rightarrow-\infty$. We have,

[TABLE]

So, $y\in\overline{W}^{u}(x;\textbf{e})$, i.e. $\overline{W}^{u}(e;\textbf{e})\subset\overline{W}^{u}(x;\textbf{e})$. Conversely, if $y\in\overline{W}^{u}(x;\textbf{e})$ then $D(\overline{f}^{i}(y),\overline{f}^{i}(x))\rightarrow 0$ as $i\rightarrow-\infty$, and

[TABLE]

So, $y\in\overline{W}^{u}(e;\textbf{e})$, i.e. $\overline{W}^{u}(x;\textbf{e})\subset\overline{W}^{u}(e;\textbf{e})$.

(3) Its proof is the same as part (2). ∎

For simplicity, let $\Gamma_{\overline{f}}=\overline{W}^{u}(e;\textbf{e})\cap\Gamma$ and $\Gamma_{\overline{A}}=\overline{L}^{u}(e)\cap\Gamma$.

Proposition 3.2**.**

The following statements hold:

1. (1)

$\Gamma_{\overline{A}}$* and $\Gamma_{\overline{f}}$ are subgroups of $\Gamma$.* 2. (2)

$\Gamma_{\overline{f}}\subset\Gamma_{\overline{A}}$. 3. (3)

$\overline{h}(v\gamma)=\overline{h}(v)\gamma$, for each $\gamma\in\Gamma_{\overline{f}}$ and $v\in\overline{W}^{u}(e;\textbf{e})$. 4. (4)

If $\overline{W}^{u}(\gamma_{1};\textbf{e})=\overline{W}^{u}(\gamma_{2};\textbf{e})$, for some $\gamma_{1},\gamma_{2}\in\Gamma$, then we have

[TABLE]

Proof.

(1) Let $\gamma_{1},\gamma_{2}\in\Gamma_{\overline{A}}$. Since $\Gamma$ is a group we have $\gamma_{1}\gamma_{2}^{-1}\in\Gamma$. Now consider that $\gamma_{1},\gamma_{2}\in\overline{L}^{u}(e)$, since $A^{i}(e)=e$, for all $i$, then by definition,

[TABLE]

As $D$ is left invariant we have

[TABLE]

Thus $\gamma_{1}\gamma_{2}^{-1}\in\overline{L}^{u}(e)$ and $\overline{L}^{u}(e)$ is a subgroup of $N$. So $\overline{L}^{u}(e)\cap\Gamma$ is a subgroup of $\Gamma$.

For the second part, Let $\gamma_{1},\gamma_{2}\in\Gamma_{\overline{f}}$. Since $\Gamma$ is a group we have $\gamma_{1}\gamma_{2}^{-1}\in\Gamma$. Now consider that $\gamma_{1},\gamma_{2}\in\overline{W}^{u}(e;\textbf{e})$. Then,

[TABLE]

Similarly, $\overline{W}^{u}(e;\textbf{e})\gamma_{2}=\overline{W}^{u}(e;\textbf{e})$. So we have $\overline{W}^{u}(e;\textbf{e})\gamma_{1}=\overline{W}^{u}(e;\textbf{e})\gamma_{2}$ and then $\gamma_{1}\gamma_{2}^{-1}\in\overline{W}^{u}(e;\textbf{e})$, and we have the result.

(2) Take $\gamma\in\Gamma_{\overline{f}}$, such that $\gamma\notin\Gamma_{\overline{A}}$. So, $\gamma\notin\overline{L}^{u}(e)$ and for each $n\in\mathbb{Z},\;n\neq 0$, $\gamma^{n}\notin\overline{L}^{u}(e)$. On the other hand, by part (1), remark 2.15 and the fact that $\overline{h}(e)=e$, for all $n\in\mathbb{Z}$, we have $\gamma^{n}\in\overline{W}^{u}(e;\textbf{e})\subset U_{\delta_{K}}(\overline{L}^{u}(e))$, which is impossible.

(3)Let $\gamma\in\Gamma_{\overline{f}}$ and $v\in\overline{W}^{u}(e;\textbf{e})$. We have

[TABLE]

so,

[TABLE]

By part (2), $\gamma\in\Gamma_{\overline{A}}$, Thus

[TABLE]

Again by Lemma 3.1 (3) and last part of the above relation, $\overline{L}^{u}(\overline{h}(v)\gamma)=\overline{L}^{u}(e)$, and

[TABLE]

On the other hand, by part (4) of lemma 2.14, $\overline{h}(v\gamma)\in\overline{L}^{s}(\overline{h}(v)\gamma)$. Since $\overline{L}^{u}(\overline{h}(v)\gamma)\cap\overline{L}^{s}(\overline{h}(v)\gamma)=\{\overline{h}(v)\gamma\}$ (see [18] Lemma 2.1), then $\overline{h}(v\gamma)=\overline{h}(v)\gamma$.

(4) Let $x\in\overline{W}^{u}(\gamma_{1};\textbf{e})=\overline{W}^{u}(\gamma_{2};\textbf{e})$. For $\gamma_{1},\gamma_{2}\in\Gamma$, we have $\gamma_{2}\in\overline{W}^{u}(\gamma_{2};\textbf{e})=\overline{W}^{u}(\gamma_{1};\textbf{e})=\overline{W}^{u}(e;\textbf{e})\gamma_{1}$. Thus, $\gamma_{2}\gamma_{1}^{-1}\in\overline{W}^{u}(e;\textbf{e})$, and then $\gamma_{2}\gamma_{1}^{-1}\in\Gamma_{\overline{f}}$. Similarly, $x\gamma_{1}^{-1},x\gamma_{2}^{-1}\in\overline{W}^{u}(e;\textbf{e})$. Now, by part (3),

[TABLE]

∎

According to part (4) of proposition 3.2, we can define a map $\overline{h}^{\prime}:\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})\rightarrow\bigcup_{\gamma\in\Gamma}\overline{L}^{u}(\gamma)$, by

[TABLE]

Next lemma shows some properties of $\overline{h}^{\prime}$:

Proposition 3.3**.**

The following statements hold:

1. (1)

$\overline{A}\circ\overline{h}^{\prime}=\overline{h}^{\prime}\circ\overline{f}$* on $\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})$,* 2. (2)

$D(\overline{h}^{\prime},id_{|\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})})<\infty$, 3. (3)

$\overline{h}^{\prime}(\gamma)=\gamma$* for $\gamma\in\Gamma$,* 4. (4)

if $x\in\overline{W}^{u}(\gamma;\textbf{e})\>(\gamma\in\Gamma)$, then $\overline{h}^{\prime}(x)\in\overline{L}^{u}(\gamma)$ and $\overline{h}^{\prime}(x)\in\overline{L}^{s}(\overline{h}(x))$, 5. (5)

if $y\in\overline{W}^{s}(x)$ for $x,y\in\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})$, then $\overline{h}^{\prime}(y)\in\overline{L}^{s}(\overline{h}^{\prime}(x))$.

Proof.

(1) Suppose that $x\in\overline{W}^{u}(\gamma;\textbf{e})=\overline{W}^{u}(e;\textbf{e})\gamma$, for some $\gamma\in\Gamma$. Then

[TABLE]

By [2] page 205, we have \overline{f}\big{(}\overline{W}^{u}(\gamma;\textbf{e})\big{)}=\overline{W}^{u}(\overline{f}(\gamma);\textbf{e}). Here $\overline{f}(\gamma)$ means $\overline{f}_{*}(\gamma)$ which by lemma 2.8 is equal to $\overline{A}_{*}(\gamma)$ and $\overline{A}_{*}(\gamma)\in\Gamma$. Therefore,

[TABLE]

so,

[TABLE]

Thus we have

[TABLE]

(2) Let $x\in\overline{W}^{u}(\gamma;\textbf{e})$, for some $\gamma\in\Gamma$, and let $\delta_{K}>0$ be satisfying $D(\overline{h},id_{N})<\delta_{K}$. Then

[TABLE]

(3) For any $\gamma\in\Gamma$, by definition we have

[TABLE]

(4) Let $x\in\overline{W}^{u}(\gamma;\textbf{e})$, for some $\gamma\in\Gamma$. We have

[TABLE]

and

[TABLE]

(5) By the second part of proof of (4), we have

[TABLE]

so, $\overline{h}^{\prime}(y)\in\overline{L}^{s}(\overline{h}^{\prime}(x))$. ∎

Lemma 3.4** ([18] Lemma 7.6).**

For each $u,v\in N$, $\overline{W}^{u}(u;\textbf{e})\cap\overline{W}^{s}(v)$ is the set of one point.

According to the above lemma, define $\overline{\iota}(u,v)=\overline{W}^{u}(u;\textbf{e})\cap\overline{W}^{s}(v)$.

Lemma 3.5**.**

For $\epsilon>0$, there is $\delta>0$ such that

[TABLE]

Proof.

Let $\epsilon>0$ be given. Since $\overline{h}$ is $D$-biuniformly contiuous there exists $\epsilon^{\prime}>0$ such that

[TABLE]

By [2] theorem 6.6.5 or [18] lemma 7.2, since $N$ is simply connected, $\overline{A}$ has local product structure (for definition and details, see [2]), and then for $\epsilon>0$ there exists $\delta^{\prime}>0$ such that

[TABLE]

Again since $\overline{h}$ is $D$-biuniformly continuous, there exists $\delta>0$ such that

[TABLE]

We know that \overline{h}\big{(}\overline{\iota}(u,v)\big{)}=\beta(\overline{h}(u),\overline{h}(v)) therefore

[TABLE]

∎

Proposition 3.6**.**

$\overline{h}^{\prime}$* is $D$-uniformly continuous.*

Proof.

Suppose that the statement is false. So there is $\epsilon_{0}>0$, for all $\delta>0$, there are $x,y\in\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})$ such that

[TABLE]

By definition of $\overline{L}^{\sigma}(x)$ $(x\in N,\;\sigma=s,u)$, for $w\in\overline{L}^{s}(v)$ there is $\epsilon_{1}>0$ such that

[TABLE]

Take $n>0$ and $\delta_{1}>0$ such that $\epsilon^{n}\geq 2\delta_{K}$ and $\delta_{1}^{n}\leq 2\delta_{K}$.

By Lemma 2.20, there exists $\delta_{2}>0$ such that

[TABLE]

Since $\overline{h}$ is continuous, take $\delta_{3}>0$ such that

[TABLE]

By lemma 3.5, there is $0<\delta<\delta_{2}$ such that

[TABLE]

Now consider $x,y\in\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})$ satisfy (3.4). There exist $\gamma_{x},\gamma_{y}\in\Gamma$ such that $x\in\overline{W}^{u}(\gamma_{x},\textbf{e})$ and $y\in\overline{W}^{u}(\gamma_{y},\textbf{e})$. We have

[TABLE]

By proposition 3.3.(4)

[TABLE]

Thus by proposition 3.3.(5), (3) and (3.5) we have

[TABLE]

Suppose $\gamma=\gamma_{y}\gamma_{x}^{-1}$. We have

[TABLE]

On the other hand,

[TABLE]

Now we have

[TABLE]

Finally, (3) and (3) make a contradiction. ∎

Let $\tilde{u}=(u_{i})\in N_{\overline{f}}$ and for each $i\in\mathbb{Z}$, $\overline{f}_{u_{i},u_{i+1}}$ be the lift of $f$ by $\pi$ such that $\overline{f}(u_{i})=u_{i+1}$ and define

[TABLE]

We define a map $\tau_{\tilde{u}}=\tau_{\tilde{u}}^{f}:N\rightarrow(N/\Gamma)_{f}$ by

[TABLE]

Since $\overline{f}(e)=e$, then $\tau_{\textbf{e}}(e)=\tau_{\tilde{e}}(e)=(\pi(e))_{i=-\infty}^{\infty}$.

Lemma 3.7** ([2] Lemma 6.6.8 (1)).**

If $x\in X$ and $\tilde{u}\in N_{f}$ then $\pi(\overline{W}^{u}(x;\tilde{u}))=W^{u}(\tau_{\tilde{u}}(x))$.

Let $X$ be a compact metric set and $f:X\rightarrow X$ a continuous surjection. A point $x\in X$ is said to be a nonwandering point if for any neighborhood $U$ of $x$ there is an integer $n>0$ such that $f^{n}(U)\cap U\neq\emptyset$. The set $\Omega(f)$ of all nonwandering points is called the nonwandering set. Clearly $\Omega(f)$ is closed in $X$ and invariant under $f$.

$f$ is said to be topologically transitive (here $X$ may be not necessarily compact), if there is $x_{0}\in X$ such that the orbit $O^{+}(x_{0})=\{f^{i}(x_{0}):i\in\mathbb{Z}^{\geq 0}\}$ is dense in $X$. It is easy to check that if $X$ is compact, a continuous surjection $f:X\rightarrow X$ is topologically transitive if and only if for any $U,V$ nonempty open sets there is $n>0$ such that $f^{n}(U)\cap V\neq\emptyset$.

A continuous surjection $f:X\rightarrow X$ of a metric space is topologically mixing if for nonempty open sets $U,V$ there exists $N>0$ such that $f^{n}(U)\cap V\neq\emptyset$ for all $n>N$. Topological mixing implies topological transitivity.

For continuing, we need next theorem for which proof one can see [2] Theorem 3.4.4.

Theorem 3.8** (Topological decomposition theorem).**

Let $f:X\rightarrow X$ be a continuous surjection of a compact metric space. If $f:X\rightarrow X$ is a $TA$-map, then the following properties hold:

1. (1)

(Spectral decomposition theorem due to Smale) The nonwandering set, $\Omega(f)$, contains a finite sequence $B_{i}\;(1\leq i\leq l)$ of $f$-invariant closed subsets such that

1. (i)

$\Omega(f)=\bigcup_{i=1}^{l}B_{i}$* (disjoint union),* 2. (ii)

$f_{|B_{i}}:B_{i}\rightarrow B_{i}$* is topologically transitive.*

Such the subsets $B_{i}$ are called basic sets. 2. (2)

(Decomposition theorem due to Bowen) For $B$ a basic set there exist $a>0$ and a finite sequence $C_{i}\;(0\leq i\leq a-1)$ of closed subsets such that

1. (i)

$C_{i}\cap C_{j}=\emptyset\;(i\neq j),\;f(C_{i})=C_{i+1}$* and $f^{a}(C_{i})=C_{i}$,* 2. (ii)

$B=\bigcup_{i=0}^{a-1}C_{i}$, 3. (iii)

$f^{a}_{|C_{i}}:C_{i}\rightarrow C_{i}$* is topologically mixing,*

Such the subsets $C_{i}$ are called elementary sets.

Lemma 3.9** ([18] Lemma 5.4).**

$\Omega(f)=N/\Gamma$.

Lemma 3.10**.**

$N/\Gamma$* is indeed an elementary set.*

Proof.

By lemma 2.10, let $\overline{f}:N\rightarrow N$ be the lift of $f$ such that $\overline{f}(e)=e$. By the commuting diagram:

[TABLE]

we have,

[TABLE]

Therefore, $[e]$ is a fixed point of $f$.

By lemma 3.9, $\Omega(f)=N/\Gamma$. As $N$ is connected and $\pi$ is a continuous surjection then $N/\Gamma$ is connected. In the proof of part (1) of spectral decomposition theorem, they prove that basic sets are close and open. Hence by connectedness of $\Omega(f)=N/\Gamma$, it consists of only one basic set, say $B$. On the other hand, by part (2) of spectral decomposition theorem, $N/\Gamma=B$ is the union of elementary sets. There is an elementary set, say $C$, such that $[e]\in C$. Since elementary sets are disjoint, by condition $f(C_{i})=C_{i+1}$, $N/\Gamma=B$ consists of only one elementary set. ∎

Lemma 3.11** ([2] Remark 5.3.2 (2)).**

Let $f:X\rightarrow X$ be a $TA$-map of a compact metric space and let $C$ be an elementary set of $f$. If $\tilde{u}=(u_{i})\in N_{f}$ and $x_{i}\in C$ for all $i\in\mathbb{Z}$ then $W^{u}(\tilde{x})\cap C$ is dense in $C$.

Lemma 3.12**.**

$\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})$* is dense in $N$.*

Proof.

By lemma 2.17 and lemma 3.7 we have

[TABLE]

We have \tau_{\textbf{e}}(e)=\big{(}\pi(e)\big{)}_{i=-\infty}^{\infty}\in(N/\Gamma)_{f}. On the other hand, Since by lemma 3.10, $\Omega(f)=N/\Gamma$ is an elementary set, say $C$, and for \big{(}\pi(e)\big{)}_{i=-\infty}^{\infty} we have $\pi(e)\in N/\Gamma=C$ for all $i\in\mathbb{Z}$, by lemma 3.11 we have

[TABLE]

is dense in $C=N/\Gamma$. By relation (3.12), we have the desired result. ∎

By lemma 3.6, $\overline{h}^{\prime}$ is extended to a continuous map $\tilde{h}:N\rightarrow N$. From proposition 3.3 (1), (2) and (3), and lemma 2.13, we have $\overline{h}=\tilde{h}$ and $\overline{h}(\gamma)=\gamma$ for all $\gamma\in\Gamma$.

Proposition 3.13**.**

For all $\gamma\in\Gamma$ and $x\in N$, $\overline{h}(x\gamma)=\overline{h}(x)\gamma$.

Proof.

According to lemma 2.14.(4), we have

[TABLE]

Suppose that $x\in\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})$. Then there is $\gamma_{x}\in\Gamma$ such that $x\in\overline{W}^{u}(\gamma_{x};\textbf{e})$. For each $\gamma\in\Gamma$ we have

[TABLE]

Thus

[TABLE]

On the other hand,

[TABLE]

By (3), we have \overline{L}^{u}(\gamma_{x}\gamma)=\overline{L}^{u}\big{(}\overline{h}(x)\gamma\big{)}. Therefore, by (3) we have

[TABLE]

By (3.13) and (3.16) we have

[TABLE]

Thus for each $x\in\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})$ we have $\overline{h}(x\gamma)=\overline{h}(x)\gamma$. Since $\overline{h}$ is continuous and $\bigcup_{\gamma\in\Gamma}\overline{W}^{u}(\gamma;\textbf{e})$ is dense in $N$, we have the desired result. ∎

The end of main theorem’s proof: According to proposition 3.13, $\overline{h}$ induces a homeomorphism $h:N/\Gamma\rightarrow N/\Gamma$ such that $h\circ\pi=\pi\circ\overline{h}$. i.e. the following diagram commutes:

[TABLE]

$h$ is the conjugacy between $f$ and $A$. For if $x\in N/\Gamma$ then there is $y\in N$ such that $x=\pi(y)$ and

[TABLE]

So the Main Theorem is proved.

Proof of Corrollary 1.11. As mentioned in section 2, every endomorphism of a compact metric space is a covering map. Every Anosov endomorphism is a $TA$-map (see [2] Theorem 1.2.1). Every diffeomorphism is special (since it is injective). For every diffepmrphism or special expanding map of a nil-manifold, by (repaired for nil-manifolds) Theorem 1.9, it is conjugate to a hyperbolic nil-automorphism or an expanding nil-endomorphism, respectively, which are hyperbolic nil-endomorphisms. In Theorem 1.10, we prove the case that $f$ is not injective or expanding. So in this case $f$ is conjugate to a hyperbolic nil-endomorphism too.