2.1 Anosov maps

There are two classical ways to study an Anosov map. One is by using the inverse limit space and the other is by observations on the universal cover space. In this subsection, we recall these two methods and some useful propositions for this paper.

We first introduce the dynamics on the inverse limit space. Let (M,d) be a compact metric space and $M^{\mathbb{Z}}:=\{(x_i) \;|\;x_i\in M, \forall i\in \mathbb{Z} \}$ be the product topological space. It is clear that $M^{\mathbb{Z}}$ is compact and metrizable by the metric

\begin{align*} \bar{d}((x_i),(y_i))= \sum_{-\infty}^{+\infty} \frac{d(x_i,y_i)}{2^{|i|}}.\end{align*}

For any $\bar{x}=(x_i)\in M^{\mathbb{Z}}$ , let $(\bar{x})_0:=x_0$ and $\pi_0:M^{\mathbb{Z}}\to M$ be the projection $\pi_0((x_i))=x_0$ . Let $\sigma: M^{\mathbb{Z}}\to M^{\mathbb{Z}}$ be the (left) shift homeomorphism $(\sigma (x_i))_j=x_{j+1}$ , for all $j\in \mathbb{Z}$ . For a continuous map $f:M\to M$ , define the inverse limit space of f as

(2.1) \begin{align} M_f:=\{(x_i) \;|\; x_i\in M \ \textrm{and} \ f(x_i)=x_{i+1}, \forall i\in \mathbb{Z} \}.\end{align}

Denote the restriction of $\sigma$ on $M_f$ by $\sigma_f$ . It is clear that the inverse limit space $(M_f,\bar{d})$ is a $\sigma_f$ -invariant and compact metric space.

Definition 2.1 [Reference PrzytyckiPrz76]. A $C^1$ local diffeomorphism $f:M\to M$ is called an Anosov map if there exist constants $C>0$ and $0<\mu<1$ such that, for every $\tilde{x}=(x_i) \in M_f$ , there exists a splitting

\begin{align*} T_{x_i}M=E_f^s(x_i,\tilde{x})\oplus E_f^u(x_i,\tilde{x}),\quad \forall i\in \mathbb{Z}, \end{align*}

which is Df-invariant so

\begin{align*} D_{x_i}f(E_f^{s}(x_i,\tilde{x})) =E_f^s(x_{i+1},\tilde{x}) \quad\textrm{and}\quad D_{x_i}f(E_f^{u}(x_i,\tilde{x})) =E_f^u(x_{i+1},\tilde{x}), \quad\forall i\in\mathbb{Z}, \end{align*}

and, for all $n>0$ , one has that

\begin{align*} &\|D_{x_i}f^n (v)\|\le C\mu^n\|v\|, \qquad\quad \forall v\in E_f^s(x_i,\tilde{x}), \;\;\forall i\in\mathbb{Z},\\ &\|D_{x_i}f^n (v)\|\ge C^{-1}\mu^{-n}\|v\|, \;\;\quad \forall v\in E_f^u(x_i,\tilde{x}),\;\;\forall i\in\mathbb{Z}. \end{align*}

Note that Definition 2.1 is equivalent to Definition 1.1 [Reference SakaiSak87]. In particular, by Definition 2.1 with the concept of cone-fields (see [Reference An, Gan, Gu and ShiAGGS23, Proposition 2.12]), the stable bundle $E^s_f$ is independent of the choice of orbit for every point. Meanwhile, the unstable direction relies on the negative orbits. We say $f\in\mathbf{A}^1(M)$ is special if the unstable directions on each point are independent of the negative orbits, i.e., there exists a Df-invariant continuous splitting $TM=E^s_f\oplus E^u_f$ such that Df is uniformly contracting in $E_f^s$ and uniformly expanding in $E^u_f$ . Note that Anosov diffeomorphisms and expanding maps are always special.

Since every Anosov map on $\mathbb{T}^d$ is transitive ([Reference Aoki and HiraideAH94], and see also [Reference An, Gan, Gu and ShiAGGS23, § 4]), it follows from [Reference PrzytyckiPrz76, Reference Micena and TahzibiMT16] that non-special Anosov maps are $C^1$ -dense non-invertible Anosov maps with non-trivial stable bundles and that each of them admits a residual set in which each point has infinitely many unstable directions. In particular, let $f\in\mathbf{A}^1(\mathbb{T}^d)$ be non-special, then f is not conjugate to its linearization $f_*:\pi_1(\mathbb{T}^d) \to \pi_1(\mathbb{T}^d)$ . In fact, one has the following proposition.

Despite all this, there exists a conjugacy on the level of inverse limit spaces for toral Anosov maps. For convenience, we state it here.

For $f\in \mathbf{N}^1(\mathbb{T}^d)$ and $x\in\mathbb{T}^d$ , define the preimage set P(x,f) of x by

(2.2) \begin{align} P(x,f)=\bigcup_{k\geq 0} P_k(x,f) \quad \textrm{where } P_k(x,f)=\{y\in\mathbb{T}^d\;|\; f^k(y)=f^k(x)\}.\end{align}

We also have the stable foliation, just as in the case of an Anosov diffeomorphism, and each leaf of this is also independent of the choice of orbits. Note that in this sense we cannot define the stable leaf of a point x by collecting points whose positive orbits are closed to one of x, since we need to exclude its k-preimage set $P_k(x,f)$ for any $k\in\mathbb{N}$ . We must therefore redefine the stable leaf in the strong sense (see [Reference Aoki and HiraideAH94, § 5.1]) by linking the local stable leaves with small size $R>0$ ,

$$\mathcal{F}_{f,R}^s(x)=\{ y\in\ M \;|\; d(f^i(x),f^i(y))<R, \; \forall i\geq 0 \}.$$

We also have the (local) unstable leaf, but this relies on the choice of negative orbits. Let $\tilde{x}=(x_i)\in M_f$ and $R>0$ . We define the local unstable leaf of $x_0$ for $\tilde{x}$ with size R by

$$\mathcal{F}_{f,R}^u(x_0,\tilde{x})=\{ y\in M\;|\; \exists(y_i)\in M_f \ \textrm{with}\ y_0=y \;\;\textrm{such that}\;\; d(x_{-i},y_{-i})<R, \; \forall i\leq0 \}.$$

It is clear that the stable leaf $\mathcal{F}_f^s(x)$ and the local unstable leaf $\mathcal{F}_{f,R}^u(x_0,\tilde{x})$ are both submanifolds and are tangent to $E^s_f(x_0)$ and $E^u_f(x_0,\tilde{x})$ , respectively (see [Reference PrzytyckiPrz76, Reference Qian, Xie and ZhuQXZ09]). Moreover, as in the case of Anosov diffeomorphisms, the local stable leaves of an Anosov map form a family of embeddings varying continuously with $x\in M$ in $C^1$ -topology. In particular, one has a similar property for local unstable leaves as follows.

Proposition 2.4 [Reference PrzytyckiPrz76]. Let $\tilde{x}^k=(x_i^k)_{i\in\mathbb{Z}}\in M_f (k\in\mathbb{N})$ be a sequence of orbits. If $\tilde{x}^k\to \tilde{y}=(y_i)\in M_f$ as $k\to+\infty$ , then for any $R>0$ ,

$$\mathcal{F}_{f,R}^u(x_0^k,\tilde{x}^k)\longrightarrow \mathcal{F}_{f,R}^u(y_0,\tilde{y}),$$

in $C^1$ -topology as $k\to+\infty$ . In particular, $E^u_f(y_0,\tilde{y})$ is continuous with respect to $(y_0,\tilde{y})\in M\times M_f$ .

We also have the Local Product Structure for Anosov maps (see [Reference Aoki and HiraideAH94, § 5.2] and [Reference Qian, Xie and ZhuQXZ09, IV.2]) as in the following proposition.

Although the inverse limit space has compactness, which the universal cover lacks, observing the dynamics on the universal cover has some added convenience. In fact, for a local diffeomorphism f on a closed Riemannian manifold M, Ma $\rm \widetilde{n} \acute{e}$ and Pugh [Reference Mañé and PughMP75] proved that it is an Anosov map if and only if its lift $F:\widetilde{M}\to \widetilde{M}$ on the universal cover is an Anosov diffeomorphism, i.e., there exist constants $C>0$ , $0<\mu<1$ and a DF-invariant splitting $T\widetilde{M}=E^s_F\oplus E^u_F$ such that for all $n>0$ and all $x\in\widetilde{M}$ , one has

\begin{align*} &\|D_{x}F^n (v)\|\le C\mu^n\|v\|, \quad \quad\quad \forall v\in E_F^s(x),\\ &\|D_{x}F^n (v)\|\ge C^{-1}\mu^{-n}\|v\|, \;\;\quad \forall v\in E_F^u(x).\end{align*}

Moreover, when we restrict to $\mathbb{T}^d$ , the lifts on $\mathbb{R}^d$ of an Anosov map and its linearization are conjugate [Reference Aoki and HiraideAH94, Proposition 8.2.1 and Proposition 8.4.2]. For convenience, we restate these as the following proposition.

In the remainder of this subsection, we always assume that $f\in\mathbf{A}^1(\mathbb{T}^d)$ and $A:=f_*:\pi_1(\mathbb{T}^d)\to\pi_1(\mathbb{T}^d)$ is its linearization. Let $F:\mathbb{R}^d \to \mathbb{R}^d$ be a lift of f by the natural projection $\pi:\mathbb{R}^d\to \mathbb{T}^d$ . For short, we denote by $A:\mathbb{R}^d\to \mathbb{R}^d$ a lift of A by $\pi$ , if there is no confusion. Recall that if $f\in\mathbf{A}^1(\mathbb{T}^d)$ , then its linearization A is in $ \mathbf{A}^1(\mathbb{T}^d)$ [Reference Aoki and HiraideAH94, Theorem 8.1.1].

For any $n\in\mathbb{Z}^d$ , let $T_n:\mathbb{R}^d\to \mathbb{R}^d$ be the deck transformations (or $\mathbb{Z}^d$ -action) $T_n(x)=x+n$ . By Proposition 2.2, one has that f is special if and only if H is commutative with deck transformations, namely,

$$H(x+n)=H(x)+n,\quad \forall x\in\mathbb{R}^d \quad\textrm{and}\quad \forall n\in\mathbb{Z}^d.$$

Recall that the stable manifolds of the lift F, denoted by $\widetilde{\mathcal{F}}_f^s(x)$ , have the following topological characterization

$$ \widetilde{\mathcal{F}}_f^s(x):=\{y\in \mathbb{R}^d \; |\; d(F^k(y),F^k(x))\to 0 \;\;\textrm{as}\;\; k\to +\infty \},$$

for all $x\in \mathbb{R}^d$ , and the unstable manifolds $\widetilde{\mathcal{F}}_f^u(x)$ by iterating backwards. Note that the two foliations $\widetilde{\mathcal{F}}_f^s$ and $\widetilde{\mathcal{F}}_f^u$ admit the Global Product Structure, namely that any two leaves $\widetilde{\mathcal{F}}_f^s(x)$ and $\widetilde{\mathcal{F}}_f^u(y)$ transversely intersect at a unique point in $\mathbb{R}^d$ .

For short, we denote the stable/unstable bundles and foliations of the linearization A on $\mathbb{T}^d$ by $L^{s/u}$ , $\mathcal{L}^{s/u}$ and on $\mathbb{R}^d$ by $\tilde{L}^{s/u}$ , $\widetilde{\mathcal{L}}^{s/u}$ , respectively. In the following, the local (un)stable manifolds with size $\delta$ of F, denoted by $\widetilde{\mathcal{F}}_f^{s/u}(x,\delta)$ , means

$$\widetilde{\mathcal{F}}_f^{s/u}(x,\delta):=\{y\in \widetilde{\mathcal{F}}_f^{s/u}(x) \; |\; d_{\widetilde{\mathcal{F}}_f^{s/u}}(x,y)\leq \delta \}, $$

for all $x\in \mathbb{R}^d$ , where $d_{\widetilde{\mathcal{F}}_f^{s/u}}(\cdot,\cdot)$ is induced by restriction of the metric on $\mathbb{R}^d$ along the leaves of $\widetilde{\mathcal{F}}_f^{s/u}$ . We also denote by $\mathcal{F}_f^s(x_0,\delta)$ and $\mathcal{F}_f^u(x_0,\tilde{x},\delta)$ the local stable and unstable leaves of f for $(x_i)=\tilde{x}\in \mathbb{T}^d_f$ with size $\delta$ measured by metrics along leaves.

It is clear that H maps the stable/unstable leaf of F to one of A, from the topological character of a stable/unstable leaf, namely,

(2.3) \begin{align} H(\widetilde{\mathcal{F}}_f^{s/u}(x))=\widetilde{\mathcal{L}}^{s/u}(H(x)),\quad \forall x\in\mathbb{R}^d.\end{align}

We denote the map in the stable leaf space of F by $F_s:\mathbb{R}^d/ \widetilde{\mathcal{F}}_f^{s} \to \mathbb{R}^d/ \widetilde{\mathcal{F}}_f^{s}$ with $F_s( \widetilde{\mathcal{F}}_f^s(x))=\widetilde{\mathcal{F}}_f^s(F(x))$ . It follows from (2.3) that H induces a conjugacy between the stable leaf spaces,

(2.4) \begin{align} H_{s}:\mathbb{R}^d/ \widetilde{\mathcal{F}}_f^{s} \to \mathbb{R}^d/ \widetilde{\mathcal{L}}^{s},\end{align}

with $H_s(\widetilde{\mathcal{F}}_f^s(x) )=\widetilde{\mathcal{L}}^s(H(x))$ such that $H_s\circ F_s= A_s\circ H_s$ .

Although in general H cannot be commutative with deck transformations, the next proposition says that $H_s$ is always commutative with deck transformations.

Proposition 2.7 [Reference An, Gan, Gu and ShiAGGS23]. Assume that H is given by Proposition 2.6. Then for any $x\in\mathbb{R}^d$ and $n\in\mathbb{Z}^d$ ,

$$H(x+n)-n\in \widetilde{\mathcal{L}}^s(H(x)) \quad \textrm{and } \quad H^{-1}(x+n)-n\in \widetilde{\mathcal{F}}_f^s(H^{-1}(x)).$$

By Proposition 2.7, one can get the following proposition, which means that H is ‘asymptotically’ commutative with the deck transformation $T_n$ , for $n\in A^m\mathbb{Z}^d$ with $m\to+\infty$ .

Proposition 2.8 [Reference An, Gan, Gu and ShiAGGS23]. Assume that H is given by Proposition 2.6. There exist $C>0$ and $\{\varepsilon_m\}$ with $\varepsilon_m\to 0$ as $m\to +\infty$ , such that for every $x\in\mathbb{R}^d$ and every $n_m\in A^m\mathbb{Z}^d$ , one has

$$|H(x+n_m)-H(x)-n_m|<C\cdot \| A|_{\widetilde{\mathcal{L}}^s}\|^m$$

and

$$|H^{-1}(x+n_m)-H^{-1}(x)-n_m|<\varepsilon_m.$$

To end this subsection, we state two properties of stable and unstable foliations of Anosov maps, whose proofs are quite similar to the case of Anosov diffeomorphisms.

The proof of this proposition uses a local analysis just like in [Reference PesinPes04, Theorem 7.1]. We mention that one may use the compactness of the dynamics of F induced by that of $\sigma_{\textit{f}}$ after projection.

Proof. We refer to [Reference Brin, Burago and IvanovBBI09] for proof of the diffeomorphism case. One can find an adaptation for the case of $\widetilde{\mathcal{F}}_f^s$ in [Reference An, Gan, Gu and ShiAGGS23]. However, since the unstable foliation $\widetilde{\mathcal{F}}_f^u$ may not be commutative with deck transformations, we will show this case. We first claim that, for any $C_1>0$ , there exists $R>0$ such that, for any $y\in\mathbb{R}^d$ and $z\in\widetilde{\mathcal{F}}_f^u(y)$ ,

(2.5) \begin{align} \textrm{if} \ \ d(y,z)<C_1 \quad \textrm{then}\ d_{\widetilde{\mathcal{F}}_f^u}(y,z)<R. \end{align}

Note that (2.5) is obvious for special Anosov maps, since via the deck transformations one can move the local leaf with endpoints y and z to the $(C_1+\sqrt{d})$ -ball at 0 in which the local leaves have a uniform size. For the coherence of the proofs of this proposition and the later Lemma 4.3, we leave the proof of this claim to Claim 4.4.

Let H be the conjugacy between F and A given by Proposition 2.6. In particular, $\|H-\textrm{ Id}_{\mathbb{R}^d}\|_{C_0}<C_0$ . Since H maps $\widetilde{\mathcal{F}}_f^{s/u}$ to $\widetilde{\mathcal{L}}^{s/u}$ , one has that $\widetilde{\mathcal{F}}_f^{s/u}(x)$ is contained in the $2C_0$ -neighborhood of $ \widetilde{\mathcal{L}}^{s/u}(x)$ , namely, $\widetilde{\mathcal{F}}_f^{s/u}(x)\subset B_{2C_0}( \widetilde{\mathcal{L}}^{s/u}(x))$ for any $x\in\mathbb{R}^d$ . Fix $n\in\mathbb{Z}^d$ such that $d(x+n, \widetilde{\mathcal{L}}^s(x))> 5C_0$ for all $x\in\mathbb{R}^d$ . Then one has

(2.6) \begin{align} d_H( \widetilde{\mathcal{L}}^s(x), \widetilde{\mathcal{F}}_f^s(x+n) )> C_0,\quad \forall x\in\mathbb{R}^d,\end{align}

where $d_H(\cdot,\cdot)$ is the Hausdorff distance.

Proof of Claim 2.12. The existence and uniqueness of the intersection point w follows from the Global Product Structure. By (2.6), we have

\[d_H(\widetilde{\mathcal{L}}^s(w), \widetilde{\mathcal{L}}^s(x))= d(w, \widetilde{\mathcal{L}}^s(x))>d_H( \widetilde{\mathcal{F}}_f^s(x+n), \widetilde{\mathcal{L}}^s(x) )>C_0.\]

Moreover, there exists $C_2>0$ depending only on $n\in\mathbb{N}$ such that $w\in B_{C_2}(\widetilde{\mathcal{L}}^s(x))$ . It follows that $w\in B_{C_2}(\widetilde{\mathcal{L}}^s(x))\bigcap B_{2C_0}(\widetilde{\mathcal{L}}^u(x))$ and hence d(w,x) has a uniform upper bound. See Figure 2. By (2.5) we get the existence of $L_0$ .

Figure 2.

Cut unstable leaf into curves with length no more than $L_0$ and $\widetilde{\mathcal{L}}^s$ -distance bigger than $C_0$ .

Now for any $x\in\mathbb{R}^d$ and $y\in\widetilde{\mathcal{F}}_f^u(x)$ , we denote $x_0=x$ . By induction and starting from $x_0$ , we apply Claim 2.12 to $x_k\ (k\geq 0)$ and get $\{x_{k+1}\}=\widetilde{\mathcal{F}}_f^u(x_k) \bigcap \widetilde{\mathcal{F}}_f^s(x_k+n)$ such that $d_H( \widetilde{\mathcal{L}}^s(x_{k+1}), \widetilde{\mathcal{L}}^s(x_k) )> C_0$ and $d_{\widetilde{\mathcal{F}}_f^u}(x_{k+1},x_k)<L_0$ . See Figure 2. Since $\widetilde{\mathcal{F}}_f^u$ is one-dimensional, each leaf can be endowed with an order $\prec$ and we can assume that $x_k\prec x_{k+1}\ (k\geq 0)$ and $x=x_0\prec y$ . Then there exists $k\geq 0$ such that $x_k\prec y \prec x_{k+1}$ or $y=x_k$ . It follows that

\[d_{\widetilde{\mathcal{F}}_f^u}(y,x)<d_{\widetilde{\mathcal{F}}_f^u}(x_0,x_{k+1})\leq (k+1)L_0 \]

and

\[ d(y,x)\geq d_H( \widetilde{\mathcal{L}}^s(y), \widetilde{\mathcal{L}}^s(x) )\geq d_H( \widetilde{\mathcal{L}}^s(x_k), \widetilde{\mathcal{L}}^s(x) )>kC_0.\]

Let $a={L_0}/{C_0}$ and $b=L_0$ . We get

(2.7) \begin{align} d_{\widetilde{\mathcal{F}}_f^{u}}(x,y)<a\cdot d(x,y)+b, \quad \forall x\in\mathbb{R}^d, \forall y\in \widetilde{\mathcal{F}}_f^u(x).\end{align}

Moreover, since $\widetilde{\mathcal{F}}_f^u$ is tangent to $E^u_F$ , which is uniformly continuous (see Proposition 2.4), the distances along the unstable leaf and in the ambient space are comparable at a small scale. Namely, there exist $C_3>1$ and ${\varepsilon}_0>0$ such that for any $x\in\mathbb{R}^d$ and $y\in\widetilde{\mathcal{F}}_f^u(x)$ , if $d(x,y)\leq {\varepsilon}_0$ , then $d_{\widetilde{\mathcal{F}}_f^u}(x,y)<C_3d(x,y)$ . Let $C=\textrm{max}\{ C_3, a+{b}/{{\varepsilon}_0}\}$ . By (2.7), we have $d_{\widetilde{\mathcal{F}}_f^{u}}(x,y)<C \cdot d(x,y)$ for any $x\in\mathbb{R}^d$ and $y\in \widetilde{\mathcal{F}}_f^{u}(x)$ .

2.2 Conjugacies of Anosov maps

In this subsection, we always assume that $f,g\in\mathbf{A}^1(\mathbb{T}^d)$ and f is homotopic to g. Without loss of generality, we can suppose $f_*=g_*:\pi_1(\mathbb{T}^d)\to\pi_1(\mathbb{T}^d)$ and $f_*$ induces $A:\mathbb{T}^d \to\mathbb{T}^d$ . We will explain the connections between the different conjugacies, that on the universal cover given by Proposition 2.6, that on the inverse limit spaces given by Proposition 2.3 and that on the stable leaf spaces given by (2.4), and also the leaf conjugacy (see (1.1)). As a corollary, we can ‘translate’ the stable periodic data condition given by the leaf conjugacy into the stable periodic condition given by conjugacies on the leaf space and inverse limit space. Here we mention that the last two ways to match the periodic data on the stable bundle can be extended to the case of dim $E^s_f>1$ .

First of all, we introduce the conjugacy on a (stable) leaf space. Here the (stable) leaf spaces for f and g are the quotient spaces $\mathbb{T}^d/ \mathcal{F}_f^s$ and $\mathbb{T}^d/ \mathcal{F}_g^s$ , respectively. For any $x\in\mathbb{T}^d$ , denote the equivalence classes of x in $\mathbb{T}^d/{\mathcal{F}_f^s}$ and $\mathbb{T}^d/{\mathcal{F}_g^s}$ by $[x]_f$ and $[x]_g$ , respectively. Then f and g induce maps on their spaces of stable leaves:

$$f_s:\mathbb{T}^d/{\mathcal{F}_f^s}\to\mathbb{T}^d/{\mathcal{F}_f^s}\quad \textrm{and} \quad g_s: \mathbb{T}^d/{\mathcal{F}_g^s}\to \mathbb{T}^d/{\mathcal{F}_g^s} \ ,$$

with $f_s([x]_f)=[f(x)]_f$ and $g_s([x]_g)=[g(x)]_g$ for any $x\in\mathbb{T}^d$ . A homeomorphism

$$s:\mathbb{T}^d/{\mathcal{F}_f^s}\to \mathbb{T}^d/{\mathcal{F}_g^s} \ $$

is called a conjugacy on (stable) leaf spaces between $f_s$ and $g_s$ if $s\circ f_s ([x]_f)= g_s\circ s([x]_f)$ for any $x\in\mathbb{T}^d$ . Using the conjugacy on leaf spaces is a natural way to connect the periodic points of f with the periodic points of g and match their periodic data. Indeed, let s be a conjugacy between $f_s$ and $g_s$ . Since s maps a periodic leaf of f to one of g and any periodic stable leaf admits a unique periodic point, s induces a map between the sets of periodic points, denoted by

(2.8) \begin{align} s^P:\textrm{Per}(f)\to \textrm{Per}(g).\end{align}

A natural way to obtain a conjugacy on leaf spaces is by projecting from the universal cover. For convenience, we collect all conjugacies given by Proposition 2.6 between all lifts of f and g by the natural projection $\pi:\mathbb{R}^d\to\mathbb{T}^d$ and denote this set by $\mathcal{H}(f,g)$ . Let $H\in\mathcal{H}(f,g)$ be the conjugacy between lifts F and G with $H\circ F=G\circ H$ . Let $H_s:\mathbb{R}^d/ \widetilde{\mathcal{F}}_f^s \to \mathbb{R}^d/ \widetilde{\mathcal{F}}_g^s$ be the conjugacy between $F_s$ and $G_s$ induced by H in the sense of (2.4) such that $H_s\circ F_s= G_s\circ H_s$ . Note that $\widetilde{\mathcal{F}}_f^s(x+n)=\widetilde{\mathcal{F}}_f^s(x)+n$ for all $x\in\mathbb{R}^d$ and all $n\in\mathbb{Z}^d$ ; moreover, we have $\pi(\widetilde{\mathcal{F}}_f^s(x))=\mathcal{F}_f^s(\pi(x))$ . Then by Proposition 2.7 we can project $H_s$ to d-torus by $\pi$ and induce a conjugacy between $f_s$ and $g_s$ . This gives us a map from $\mathcal{H}(f,g)$ to

$$\mathcal{S}^*(f,g):=\{ s: \mathbb{T}^d/{\mathcal{F}_f^s}\to \mathbb{T}^d/{\mathcal{F}_g^s}\; \;\textrm{homeomorphism}\ |\; s\circ f_s=g_s\circ s \},$$

the space of conjugacies on leaf spaces. Namely, the map

(2.9) \begin{align} \mathcal{S}_{f,g}:\mathcal{H}(f,g) \longrightarrow \mathcal{S}^*(f,g), \end{align}

such that $\mathcal{S}_{f,g}(H)([x]_f):=\pi \circ H_s( \pi^{-1} [x]_f)$ , is well defined.

We will give a restriction called ‘at bounded distance from Id’ to the conjugacies on leaf space. We will see that this condition is similar to being homotopic to Id $_{\mathbb{T}^d}$ if one observes it on the universal cover space $\mathbb{R}^d$ .

To make this statement precise, we need to mark lifts by fixed points. Indeed, any lift F of f by $\pi$ has exactly one fixed point (see [Reference FranksFra69] and [Reference Aoki and HiraideAH94, Theorem 8.1.3]). We use the notation $(F,p_0)$ to mark the unique fixed point $p_0$ of a given lift F. It follows from the Global Product Structure that the quotient space $\mathbb{R}^d/{\widetilde{\mathcal{F}}_f^s}$ is homeomorphic to $\widetilde{\mathcal{F}}_f^u(p_0)$ . Since the lift $(F,p_0)$ induces a homeomorphism $F_s:\mathbb{R}^d/{\widetilde{\mathcal{F}}_f^s}\to \mathbb{R}^d/{\widetilde{\mathcal{F}}_f^s}$ , and in order not to introduce too many symbols, by a slight abuse of notation we still denote by $F_s$ the map

$$F_s:\widetilde{\mathcal{F}}_f^u(p_0)\to \widetilde{\mathcal{F}}_f^u(p_0).$$

Let $x\in\mathbb{R}^d$ . Denote the equivalence class of x in $\mathbb{R}^d/{\widetilde{\mathcal{F}}_f^s}$ by $[x]_F$ and let $x^u=\widetilde{\mathcal{F}}_f^s(x)\cap\widetilde{\mathcal{F}}_f^u(p_0)$ . Note that $F_s(x^u)=F(x^u)=(F(x))^u$ , which means that $F_s$ is in fact the restriction of F on $\widetilde{\mathcal{F}}_f^u(p_0)$ . Define a metric $d_F(\cdot,\cdot)$ on $\mathbb{R}^d/{\widetilde{\mathcal{F}}_f^s}$ by

$$d_F([x]_F,[y]_F):=d(x^u,y^u),$$

where $d(\cdot,\cdot)$ is the Euclidean metric on $\mathbb{R}^d$ .

Let $s\in \mathcal{S}^*(f,g)$ . Then $s^P$ given by (2.8) maps a fixed point of f to one of g. Let $p\in\textrm{Fix}(f)$ and $q\in{\rm Fix}(g)$ with $s([p]_f)=[q]_g$ . Fix $p_0\in \pi^{-1}(p)$ and $q_0\in\pi^{-1}(q)$ . Then there are lifts F and G such that $p_0$ and $q_0$ are corresponding fixed points, respectively. It is clear that s can be lifted to the universal cover spaces $\mathbb{R}^d/{\widetilde{\mathcal{F}}_f^s}$ and $\mathbb{R}^d/{\widetilde{\mathcal{F}}_g^s}$ given by $(F,p_0)$ and $(G,q_0)$ ,

$$\tilde{s}_{p_0,q_0}:\mathbb{R}^d/{\widetilde{\mathcal{F}}_f^s}\to \mathbb{R}^d/{\widetilde{\mathcal{F}}_g^s}.$$

For short, we denote $\tilde{s}_{p_0,q_0}$ by $\tilde{s}$ , if there is no confusion. Again by the Global Product Structure, we can consider $\tilde{s}$ as $\tilde{s}:\widetilde{\mathcal{F}}_f^u(p_0) \to \widetilde{\mathcal{F}}_g^u(q_0)$ .

We say that $\tilde{s}$ is at bounded distance from Id if there exists $C>0$ such that for any $x\in\mathbb{R}^d$ ,

$$d_G([x]_G, \tilde{s}([x]_F))<C.$$

One can see in the following proposition that this is equivalent to the existence of a constant $C>0$ such that

$$d(x, \tilde{s}(x))<C, \ \ \forall x\in \widetilde{\mathcal{F}}_f^u(p_0).$$

We mention in advance that this restriction for s is equivalent to the fact that the corresponding leaf conjugacy h is homotopic to Id $_{\mathbb{T}^d}$ (see Remark 2.16).

Define the set $\mathcal{S}(f,g)$ by

\begin{align*} \mathcal{S}(f,g):= \{ s: \mathbb{T}^d/{\mathcal{F}_f^s}\to \mathbb{T}^d/{\mathcal{F}_g^s}\; \;\textrm{ homeomorphism} \ |\;& s\circ f_s=g_s\circ s\\ &\textrm{and}\;\; \exists\;\tilde{s} {\rm\ at \ bounded \ distance\ from \ Id} \}.\end{align*}

The following proposition says that the map $\mathcal{S}_{f,g}$ is actually a surjection from $\mathcal{H}(f,g)$ to $\mathcal{S}(f,g)$ .

See the following commutative diagram (Figure 3).

Figure 3.

The map $\mathcal{S}_{f,g}:\mathcal{H}(f,g) \to \mathcal{S}^*(f,g)$ is surjective.

By Proposition 2.6, there exists $C_1>0$ such that $\|H-\mathrm{Id}_{\mathbb{R}^d}\|_{C^0}<C_1$ . Moreover, we claim that there exists $C_0>0$ such that

(2.11) \begin{align} \widetilde{\mathcal{F}}^{s/u}_{\sigma}(x)\subset B_{C_0}(\widetilde{\mathcal{L}}^{s/u}(x)) ,\quad \forall x\in\mathbb{R}^d \quad \textrm{and }\quad \sigma=f,g, \end{align}

where $B_{C_0}(\mathcal{L})$ is the $C_0$ neighborhood of the set $\mathcal{L}$ . We just need to prove the claim for $\widetilde{\mathcal{F}}_f^s$ . Indeed, let $H_F$ be the conjugacy between F and A given by Proposition 2.6 and take $C_2$ such that $\|H_F-\mathrm{Id}_{\mathbb{R}^d}\|_{C^0}<C_2$ . Then for any $y\in\widetilde{\mathcal{F}}_f^s(x)$ one has that $y\in B_{C_2}( \widetilde{\mathcal{L}}^s( H_F(x) ))$ . It follows that $y\in B_{2C_2}( \widetilde{\mathcal{L}}^s( x))$ and we can take $C_0=2C_2$ .

By (2.11), one has $\widetilde{\mathcal{F}}_g^u(q_0)\subset B_{C_0}(\widetilde{\mathcal{L}}^u(q_0))$ and $\widetilde{\mathcal{F}}_g^s( H(x))\subset B_{C_1}( \widetilde{\mathcal{F}}_f^s(x) ) \subset B_{C_0+C_1}(\widetilde{\mathcal{L}}^s(x)) $ for any $x\in\mathbb{R}^d$ . Then we have that

$$x^u:=\ \widetilde{\mathcal{F}}_g^u(q_0) \cap\widetilde{\mathcal{F}}_g^s(x) \in B_{C_0}(\widetilde{\mathcal{L}}^u(q_0)) \cap B_{C_0+C_1}(\widetilde{\mathcal{L}}^s(x)),$$

and

$$H(x)^u:=\ \widetilde{\mathcal{F}}_g^u(q_0)\cap \widetilde{\mathcal{F}}_g^s( H(x)) \in B_{C_0}(\widetilde{\mathcal{L}}^u(q_0)) \cap B_{C_0+C_1}(\widetilde{\mathcal{L}}^s(x)).$$

See Figure 4. It follows that $d_G([x]_G, H^s([x]_F))= d(x^u, H(x)^u)$ is uniformly bounded for all $x\in\mathbb{R}^d$ , and hence $H_s$ is at bounded distance from Id.

Figure 4.

The points $H(x)^u$ and $x^u$ in the dashed box, which has uniform size.

Now we prove the second item. Let $s\in\mathcal{S}(f,g)$ and $s^P$ be given by (2.8). By the definition of $\mathcal{S}(f,g)$ , there exist lifts $(F,p_0)$ and $(G,q_0)$ of f and g with $s^P(\pi(p_0))=\pi(q_0)\in\textrm{ Fix}(g)$ such that the lift $\tilde{s}$ of s corresponding to $(F,p_0)$ and $(G,q_0)$ is at bounded distance from Id. For short, let $p=\pi(p_0)$ and $q=\pi(q_0)$ .

Let H be the conjugacy between F and G given by Proposition 2.6. By the first item, we already have $\mathcal{S}_{f,g}(H)\in\mathcal{S}(f,g)$ . Hence it suffices to prove that the conjugacy which is at bounded distance from Id between $F_s:\mathbb{R}^d/\widetilde{\mathcal{F}}_f^s\to \mathbb{R}^d/\widetilde{\mathcal{F}}_f^s$ and $G_s:\mathbb{R}^d/\widetilde{\mathcal{F}}_g^s\to \mathbb{R}^d/\widetilde{\mathcal{F}}_g^s$ is unique and hence $\mathcal{S}_{f,g}(H)=s$ .

Indeed, assume that there exist two different conjugacies $h_i:\mathbb{R}^d/\widetilde{\mathcal{F}}_f^s\to \mathbb{R}^d/\widetilde{\mathcal{F}}_g^s\ (i=1,2)$ and that both of them are at bounded distance from Id. Then there exists $x\in\mathbb{R}^d$ such that $h_1([x]_F)\neq h_2([x]_F)$ and this implies that these two leaves intersect $\widetilde{\mathcal{F}}_g^u(q_0)$ at two distinct points $y_1$ and $y_2$ . Note that for any $n\in\mathbb{Z}$ , one has

$$G^ny_i=h_i( [F^nx]_F )\cap \widetilde{\mathcal{F}}_g^u(q_0), \quad i=1,2.$$

Let $H_G$ be the conjugacy between $(G,q_0)$ and (A,0). Then $H_G(y_1)\in \widetilde{\mathcal{L}}^u( H_G(y_2) )$ and

$$d( H_G(G^ny_1),H_G(G^ny_2) )= d( A^n\circ H_G(y_1), A^n\circ H_G(y_2) ) \to +\infty, $$

as $n\to + \infty$ . It follows that $d(G^ny_1,G^ny_2 )\to +\infty$ since $d_{C^0}(H_G,\mathrm{Id}_{\mathbb{R}^d})$ is bounded. Note that the assumption that $h_1$ and $h_2$ are both at bounded distance from Id implies that both $G^ny_1$ and $G^ny_2$ have bounded distance with $[F^nx]_G\cap \widetilde{\mathcal{F}}_g^u(q_0)$ , for any $n\in\mathbb{N}$ . This is a contradiction.

Recall that a (stable) leaf conjugacy $h\in\textrm{Home}_0(\mathbb{T}^d)$ between f and g maps stable leaves of f to stable leaves of g such that $h\circ f(\mathcal{F}_f^s(x))= g \circ h (\mathcal{F}_f^s(x))$ for every $x\in\mathbb{T}^d$ . Let $f,g\in\mathbf{A}^1_1(\mathbb{T}^d)$ be homotopic. The next proposition actually says that there exists leaf conjugacy between f and g.

1. (i) Let $H_n(x) = H(x-n)+n$ for all $n\in \mathbb{Z}^d$ and $x\in \mathbb{R}^d$ . Then by Proposition 2.7, for any $k>0$ ,

   \begin{align*} \overline{H}_k:=\sum_{n\in C_k}\frac{1}{(2k+1)^d}H_n, \end{align*}
   where $C_k=\{(k_1,\ldots,k_d)\in \mathbb{Z}^d | \textrm{max} \{ |k_1|,\ldots,|k_d|\}\le k \}$ is well defined on each stable leaf.

2. (ii) $\{\overline{H}_k\}_{k\in \mathbb{N}}$ is uniformly equicontinuous and is at uniform distance from Id $_{\mathbb{R}^d}$ ; note that these two properties are inherited from $\{ H_n \}_{n\in \mathbb{Z}^d}$ . By the Arzelà–Ascoli theorem, we obtain an accumulation point of $\{\overline{H}_k\}_{k\in \mathbb{N}}$ , denoted by $\tilde{H}$ . For short, let $\overline{H}_k\to \tilde{H}$ as $k\to +\infty$ .

3. (iii) $\tilde{H}$ is a leaf conjugacy between F and A which is at uniform distance from Id $_{\mathbb{R}^d}$ and satisfies $\tilde{H}(x+n)=\tilde{H}(x)+n$ for all $n\in \mathbb{Z}^d$ . Moreover, $\tilde{H}$ is injective. Descend $\tilde{H}$ to $h:\mathbb{T}^d\to\mathbb{T}^d$ , then $h\in \textrm{Home}_0(\mathbb{T}^d)$ is a leaf conjugacy.

We note that the proof of the third item, especially the injection $\tilde{H}$ , relies on the one-dimensional stable bundle. Indeed, let $x\in \mathbb{R}^d$ and $y\in \widetilde{\mathcal{F}}_f^s(x)$ with $d(x,y)=\delta>0$ . Since H is uniformly bi-continuous and preserves the stable leaves, there exists ${\varepsilon}={\varepsilon}(\delta)>0$ such that for any $n\in\mathbb{N}$ , one has

$$ d_{\widetilde{\mathcal{L}}^s}(H_n(x),H_n(y) )=d_{\widetilde{\mathcal{L}}^s}(H(x-n),H(y-n) )>{\varepsilon}.$$

If dim $E_f^s=1$ , we can give each leaf of $\widetilde{\mathcal{F}}_f^s$ and $\widetilde{\mathcal{L}}^s$ an order $\prec$ and assume that H preserves this order, i.e., the orientation of these leaves. In particular, if $y\prec x$ , then $H_n(y)\prec H_n(x)$ . It follows that

\[d_{\widetilde{\mathcal{L}}^s}(\overline{H}_k(x), \overline{H}_k(y))=\frac{1}{(2k+1)^d}\sum_{n\in C_k}d_{\widetilde{\mathcal{L}}^s}(H_n(x),H_n(y))\geq \frac{1}{(2k+1)^d}\sum_{n\in C_k}{\varepsilon}={\varepsilon}. \]

Since $\overline{H}_k\to \tilde{H}$ as $k\to +\infty$ , one has ${\varepsilon} \leq d_{\widetilde{\mathcal{L}}^s}(\overline{H}_k(x), \overline{H}_k(y))\to d_{\widetilde{\mathcal{L}}^s}(\tilde{H}(x), \tilde{H}(y))=d( \tilde{H}(x), \tilde{H}(y) )$ . Hence $ \tilde{H}(x)\neq \tilde{H}(y )$ and $\tilde{H}$ is injective.

Remark 2.16. Through the conjugacy on the stable leaf spaces, we redefine the fact that $f,g\in\mathbf{A}^1_1(\mathbb{T}^d)$ admit the same stable periodic data by matching periodic points via some $s\in\mathcal{S}(f,g)$ , namely,

(2.12) \begin{align} \lambda^s_f(p)=\lambda^s_g( s^P(p) ), \quad \forall p\in\textrm{Per}(f), \end{align}

where $s^P$ is given by (2.8). This is equivalent to the definition given by leaf conjugacy (see (1.3)). Indeed, on the one hand, let $h\in \textrm{Home}_0(\mathbb{T}^d)$ be a leaf conjugacy between f and g given by (1.3) and $H\in \mathcal{H}(f,g)$ be a conjugacy on $\mathbb{R}^d$ corresponding to h. Then H induces $s=\mathcal{S}_{f,g}(H)\in \mathcal{S}(f,g)$ by Proposition 2.13 and it follows that s satisfies (2.12). On the other hand, let $s\in\mathcal{S}(f,g)$ satisfy (2.12). Then s induces a conjugacy $H\in\mathcal{H}(f,g)$ by the second item of Proposition 2.13 and H induces a leaf conjugacy $h\in \textrm{Home}_0(\mathbb{T}^d)$ given by Proposition 2.14. It is clear that h satisfies (1.3).

The following proposition says that every conjugacy $H\in \mathcal{H}(f,g)$ on $\mathbb{R}^d$ can induce a conjugacy on inverse limit spaces. This will be helpful for ‘translating’ the condition of matching stable periodic data on the leaf space into that on the inverse limit space (see Corollary 2.20).

Proposition 2.17. Let $f,g\in \mathbf{A}^1(\mathbb{T}^d)$ be homotopic and F,G be any two lifts of f,g by $\pi$ . Assume that $H:\mathbb{R}^d\to \mathbb{R}^d$ is the conjugacy given by Proposition 2.6 between F and G with $H\circ F=G\circ H$ . Then there exists a conjugacy $\bar{h}:\mathbb{T}^d_f\to \mathbb{T}^d_g$ between $\sigma_f:\mathbb{T}^d_f\to\mathbb{T}^d_f$ and $\sigma_g:\mathbb{T}^d_g\to\mathbb{T}^d_g$ such that $\bar{h}\circ\sigma_f=\sigma_g\circ \bar{h}$ and

(2.13) \begin{align} \bar{h}\circ\pi( \textrm{Orb}_F(x) ) = \pi( {\rm Orb}_G(H(x)) ),\ \ \forall x\in\mathbb{R}^d. \end{align}

See the following commutative diagram (Figure 5), where $\pi_0:\mathbb{T}^d_{f/g}\to \mathbb{T}^d$ is the projection $\pi_0((x_i))=x_0$ .

Figure 5.

The conjugacy H induces a conjugacy $\bar{h}$ on the inverse limit space.

Proof. Let $\mathbb{R}^d_F$ be the orbit space of $F:\mathbb{R}^d\to \mathbb{R}^d$ and $\sigma_F:\mathbb{R}^d_F\to\mathbb{R}^d_F$ be the (left) shift homeomorphism just as in (2.1), and define $\sigma_G:\mathbb{R}^d_G\to\mathbb{R}^d_G$ in the same way. Then H induces a conjugacy $\overline{H}: \mathbb{R}^d_F\to \mathbb{R}^d_G$ between the shift maps $\sigma_F$ and $\sigma_G$ such that $\overline{H}\circ \sigma_F= \sigma_G\circ \overline{H}$ and $\overline{H}(\textrm{Orb}_F(x))= \textrm{Orb}_G(H(x))$ for all $x\in\mathbb{R}^d$ . It is clear that

(2.14) \begin{align} \mathbb{T}^d_f=\overline{\bigcup_{x\in\mathbb{R}^d} \pi ( \textrm{Orb}_F(x) )}.\end{align}

Indeed, for any $\tilde{x}=(x_i)\in \mathbb{T}^d_f$ and $k\leq 0$ , let $y_k$ be a lift of $x_k$ . Then the projection of the F-orbit of $F^k(y_k)$ approaches $\tilde{x}$ . If needed, we refer to [Reference Micena and TahzibiMT16, Proposition 2.5] for more details of (2.14).

We are going to prove that $\overline{H}$ can descend to $\mathbb{T}^d_f$ by the following lemma. Recall that the metric $\bar{d}(\cdot,\cdot)$ of the inverse limit space $\mathbb{T}^d_f$ is given by $\bar{d}(\bar{x},\bar{y})= \sum_{-\infty}^{+\infty} ({d(x_i,y_i)}/{2^{|i|}})$ , for all $\bar{x}=(x_i)$ and $\bar{y}=(y_i)\in \mathbb{T}^d_f$ .

Lemma 2.18. For any orbit $\bar{x}\in\mathbb{T}^d_f$ and any two sequences $\{y_k\}$ and $\{z_k\}$ with $\pi( \textrm{Orb}_F(y_k) ) \to \bar{x}$ and $\pi( \textrm{Orb}_F(z_k) ) \to \bar{x}$ as $k\to +\infty$ , one has

$$\bar{d} ( \pi( \textrm{Orb}_G(H(y_k)) ) , \; \pi ( \textrm{Orb}_G(H(z_k))) )\to 0. $$

Proof of Lemma 2.18. It is clear that for any ${\varepsilon}>0$ , there exists $\alpha>0$ such that for any $x,y\in\mathbb{R}^d$ with $d(x,y)<\alpha$ , one has

(2.15) \begin{align} \bar{d} ( \pi( \textrm{Orb}_G(x) ) , \; \pi ( {\rm Orb}_G(y)) ) <\frac{{\varepsilon}}{2}.\end{align}

For any ${\varepsilon}>0$ , there exists $I_0\in\mathbb{N}$ such that for all $x\in\mathbb{R}^d$ , all $i\geq I_0$ and all $n\in A^i\mathbb{Z}^d$ , one has

(2.16) \begin{align} \bar{d} ( \pi( \textrm{Orb}_G(x) )\; , \; \pi ( \textrm{Orb}_G(x+n)) ) <\frac{{\varepsilon}}{2}. \end{align}

Indeed, just note that $\pi(G^j(x+n))=\pi(G^jx)$ , for any $n\in A^i\mathbb{Z}^d$ and $-i\leq j <+\infty$ .

By Proposition 2.8, for given $\alpha>0$ , there exists $I_1\in\mathbb{N}$ such that for any $x\in\mathbb{R}^d$ , $i\geq I_1$ and $n\in A^i\mathbb{Z}^d$ , one has

(2.17) \begin{align} d( H(x+n),\; H(x)+n )<\frac{\alpha}{2}.\end{align}

By the uniform continuity of H, there exists $\delta>0$ such that if $d(x,y)<\delta$ , one has

(2.18) \begin{align} d(H(x),H(y))<\frac{\alpha}{2}.\end{align}

Now fix ${\varepsilon}>0$ , then we can fix $\alpha>0$ , $\delta>0$ and $i_0\geq\textrm{max}\{I_0,I_1\}$ . The conditions that both $\pi( \textrm{Orb}_F(y_k) ) $ and $\pi( \textrm{Orb}_F(z_k) ) $ converge to $ \bar{x}$ means that there exists $N_0\in\mathbb{N}$ such that for every $k\geq N_0$ , there exists $n_k\in\mathbb{Z}^d$ satisfying

$$d(F^j(F^{-i_0}z_k+n_k ),\; F^j\circ F^{-i_0}(y_k))<\delta$$

for any $0\leq j\leq 2i_0$ . In particular, one has $d(z_k+ A^{i_0}n_k \; ,\; y_k)<\delta$ and by (2.18)

(2.19) \begin{align} d(H(z_k+ A^{i_0}n_k)\; , \; H(y_k))<\frac{\alpha}{2}.\end{align}

Since $i_0\geq I_1$ , it follows from (2.17) that $d(H(z_k+ A^{i_0}n_k) , H(z_k)+ A^{i_0}n_k) < {\alpha}/{2}$ . Combining this with (2.19), one has

$$d( H(z_k)+ A^{i_0}n_k \; , \; H(y_k) )<\alpha.$$

Then by (2.15) we get

(2.20) \begin{align} \bar{d} ( \pi(\textrm{Orb }_G(H(z_k)+A^{i_0}n_k)),\; \pi( \textrm{Orb}_G(H(y_k)) ) )<\frac{{\varepsilon}}{2}. \end{align}

On the other hand, since $i_0\geq I_0$ , by (2.16), we have

(2.21) \begin{align} \bar{d} ( \pi(\textrm{Orb }_G(H(z_k)+A^{i_0}n_k)),\; \pi( \textrm{Orb}_G(H(z_k)) ) )<\frac{{\varepsilon}}{2}.\end{align}

It follows from (2.20) and (2.21) that for any ${\varepsilon}>0$ , there exists $N_0\in\mathbb{N}$ such that if $k\geq N_0$ , then

$$\bar{d} ( \pi(\textrm{Orb }_G(H(y_k))),\; \pi( \textrm{Orb}_G(H(z_k)) ) )<{\varepsilon}. $$

This completes the proof of the lemma.

Remark 2.19. Lemma 2.18 also implies that the sequence $\{ \pi( \textrm{ Orb}_G(H(y_k)))\}$ is convergent in $\mathbb{T}^d_g$ , as $\pi( \textrm{Orb}_F(y_k) ) \to \bar{x}$ . Indeed, using the same notation as in the proof of Lemma 2.18, for every ${\varepsilon}>0$ , every $i_0\geq {\rm max}\{I_0,I_1\}$ and every Cauchy sequence $\{\pi( {\rm Orb}_F(y_k) )\}\subset \mathbb{T}^d_f$ , there exists $N>0$ such that for any $k,m>N$ , there exists $n\in\mathbb{Z}^d$ with

$$d(F^j\circ F^{-i_0}(y_k),\; F^j(F^{-i_0}y_m+n) )<\delta,$$

for all $0\leq j\leq 2i_0$ . Then one has

$$\bar{d}( \pi\circ \textrm{Orb}_G(H(y_m)),\; \pi\circ \textrm{Orb}_G(H(y_k)) )< {\varepsilon}.$$

So $\{ \pi\circ \textrm{Orb}_G(H(y_k)) \}$ is a Cauchy sequence and hence convergent in the compact metric space $(\mathbb{T}^d_g,\bar{d})$ .

We continue to prove Proposition 2.17. For any $\bar{x}\in\mathbb{T}^d_f$ , there exists $\{y_k\}\subset\mathbb{R}^d$ such that $\pi( \textrm{Orb}_F(y_k) )$ converges to $ \bar{x}$ . By Lemma 2.18, we can define $\bar{h}: \mathbb{T}^d_f\to \mathbb{T}^d_g$ as

(2.22) \begin{align} \bar{h}(\bar{x})= \lim_{k\to\infty}\pi( \textrm{Orb }_G(H(y_k))).\end{align}

By the proof of Lemma 2.18, we also get that $\bar{h}$ is a continuous.

Similarly, for any $\bar{x}\in \mathbb{T}^d_g$ and $\{z_k\}\subset \mathbb{R}^d$ with $\pi( \textrm{Orb}_G(z_k) )$ converging to $ \bar{x}$ , let $\bar{h}^{-1}$ be defined by

$$\bar{h}^{-1}(\bar{x})= \lim_{n\to\infty}\pi( \textrm{Orb }_F(H^{-1}(z_k))). $$

This is well defined and also continuous. Note that $\bar{h}\circ\bar{h}^{-1}=\mathrm{Id}_{\mathbb{T}^d_g}$ and $\bar{h}^{-1}\circ\bar{h}=\mathrm{Id}_{\mathbb{T}^d_f}$ by the definition, so we have that $\bar{h}$ is a homeomorphism. Recall that $y_k$ satisfies (2.22). Hence

\begin{align*} \sigma_g\circ \bar{h} (\bar{x}) &= \sigma_g \big(\lim_{k\to\infty} \pi \circ \textrm{Orb}_G(H(y_k))\big),\\ &= \lim_{k\to\infty} ( g \circ \pi \circ G^i(H(y_k)) )_{i\in\mathbb{Z}},\\ &= \lim_{k\to\infty} ( \pi \circ G \circ G^i(H(y_k)) )_{i\in\mathbb{Z}},\\ &= \lim_{k\to\infty} ( \pi \circ G^i(H(F(y_k))) )_{i\in\mathbb{Z}},\\ &= \lim_{k\to\infty} \pi \circ \textrm{Orb}_G(H(F(y_k))). \end{align*}

Since $\pi( \textrm{Orb}_F(y_k) )\to \bar{x}$ , we have $\pi( \textrm{Orb}_F(F(y_k)) )\to \sigma_f (\bar{x})$ . Replacing $\bar{x}$ and $y_k$ of (2.22) by $\sigma_f (\bar{x})$ and $F(y_k)$ , respectively, we get

$$\sigma_g\circ \bar{h} (\bar{x})=\lim_{k\to\infty} \pi \circ \textrm{Orb}_G(H(F(y_k)))=\bar{h}\circ \sigma_f(\bar{x}).$$

This completes the proof of the proposition.

Let $\bar{h}:\mathbb{T}^d_f\to\mathbb{T}^d_g$ be a conjugacy between $\sigma_f:\mathbb{T}^d_f\to\mathbb{T}^d_f$ and $\sigma_g:\mathbb{T}^d_g\to\mathbb{T}^d_g$ . For any $p\in\textrm{Per}(f)$ , we denote the periodic f-orbit of p by $\bar{p}\in\mathbb{T}^d_f$ . It is clear that $\bar{h}(\bar{p})$ is a periodic point of $\sigma_g:\mathbb{T}^d_g\to\mathbb{T}^d_g$ with the same period as $\bar{p}$ . We further assume that $f\in\mathbf{A}^1_1(\mathbb{T}^d)$ , and define the stable Lyapunov exponent for $\bar{p}\in\textrm{Per}(\sigma_f)$ by

$$\lambda^s(f,\bar{p}):=\lambda^s_f(\pi_0(\bar{p})) ,$$

where $\lambda^s_f(x)$ is the stable Lyapunov exponent of x for f.

Corollary 2.20. Let $f,g\in\mathbf{A}^1_1(\mathbb{T}^d)$ be homotopic. If f and g admit the same periodic data on the stable bundles, then there exist lifts F and G of f and g by $\pi$ such that the conjugacy $\bar{h}:\mathbb{T}^d_f \to \mathbb{T}^d_g$ between $\sigma_f$ and $\sigma_g$ given by Proposition 2.17 satisfies

$$\lambda^s(f,\bar{p})=\lambda^s(g,\bar{h}(\bar{p})), \quad \forall \bar{p}\in \textrm{Per}(\sigma_f).$$

Since $\bar{h}(\bar{p})\in\textrm{Per}(\sigma_g)$ and every periodic stable leaf admits exactly one periodic point, it suffices to prove that

$$\mathcal{F}_g^s((\bar{h}(\bar{p}))_0)=\mathcal{F}_g^s( s^P(p)).$$

Indeed, let $p_n\in\mathbb{R}^d$ with $\pi(p_n)=p$ such that $\pi\circ\textrm{Orb}_F(p_n)\to \bar{p}$ as $n\to +\infty$ . Then by (2.13) or (2.22), one has

$$\bar{h}(\bar{p})= \lim_{n\to +\infty} \pi\circ \textrm{ Orb}_G(H(p_n)).$$

By the second item of Proposition 2.13 and 2.10, one has

$$[\pi (H(p_n))]_g=\pi[ H(p_n) ]_G=s\circ\pi ([p_n]_F)= s([p]_f)=[s^P(p)]_g.$$

Therefore,

\begin{align*}[(\bar{h}(\bar{p}))_0]_g=\lim_{n\to+\infty}[\pi(H(p_n))]_g=[s^P(p)]_g.\\[-35pt]\end{align*}

2.3 Livschitz Theorem on the inverse limit space

In this subsection, we find a Livschitz Theorem for shift homeomorphisms on the inverse limit spaces of Anosov maps. This will be helpful for constructing an affine structure in § 3. Although we already have the Livschitz Theorem for Anosov maps (see Theorem 2.21), we can only use the Livschitz Theorem for the inverse limit space on the assumption of Corollary 2.20. Recall that M is a $C^{\infty}$ -smooth closed Riemannian manifold.

Theorem 2.21 (Livschitz Theorem for Anosov maps [Reference Katok and HasselblattKH95]). Let $f\in\mathbf{A}^{1+\alpha}(M)$ be transitive and $\psi_1, \psi_2 :M\to \mathbb{R}$ be two Hölder functions with

\begin{align*}\sum_{i=1}^{\pi({p})}\psi_1(f^i({p}))= \sum_{i=1}^{\pi({p})}\psi_2(f^i({p})), \quad \forall p\in \textrm{Per}(f),\end{align*}

where $\pi({p})$ is the period of p. Then there exists a Hölder function $v:M\to \mathbb{R}$ such that

$$\psi_1(x)=\psi_2(x)+v\circ f(x)-v(x),\quad \forall x\in M.$$

As stated in [Reference An, Gan, Gu and ShiAGGS23, § 4], the proof is quite similar to the proof of the Livschitz Theorem for transitive Anosov diffeomorphisms, which can be found in [Reference Katok and HasselblattKH95, Corollary 6.4.17 and Theorem 19.2.1]. Note that we still have the Local Product Structure (see Proposition 2.5) which is useful for obtaining the exponential Anosov closing lemma (see [Reference Katok and HasselblattKH95, Corollary 6.4.17]) and hence the Livschitz Theorem for Anosov maps.

Theorem 2.22 (Livschitz Theorem on the inverse limit space). Let $f\in\mathbf{A}^{1+\alpha}(M)$ be transitive and $\sigma_f:M_f\to M_f$ be the shift homeomorphism on the inverse limit space. Assume that ${\varphi}_1, {\varphi}_2 :M_f\to \mathbb{R}$ are two Hölder functions with

$$\sum_{i=1}^{\pi(\bar{p})}{\varphi}_1(\sigma_f^i(\bar{p}))= \sum_{i=1}^{\pi(\bar{p})}{\varphi}_2(\sigma_f^i(\bar{p})), \quad \forall \bar{p}\in \textrm{Per}(\sigma_f),$$

where $\pi(\bar{p})$ is the period of $\bar{p}$ . Then ${\varphi}_1$ is Hölder-cohomologous to ${\varphi}_2$ , i.e., there exists a Hölder function $u:M_f\to \mathbb{R}$ such that

$${\varphi}_1(\bar{x})={\varphi}_2(\bar{x})+u\circ \sigma_f(\bar{x})-u(\bar{x}),\quad \forall \bar{x}\in M_f.$$

Since we already have Theorem 2.21, we just need to reduce Theorem 2.22 to the above case. This can be done by the following lemma. Briefly, it says that a function on the inverse limit space can be made cohomologous to a function on the base manifold by ‘cutting’ the backward orbit. We mention that this method originated from the case of the shift homeomorphism of finite type in [Reference BowenBow75, § 1]. However, our case is not discrete and we need a more delicate calculation with the help of the Local Product Structure (see Proposition 2.5). Note that this lemma only needs f to be $C^{1}$ -smooth.

Proof. For any $x\in M$ , we fix an orbit $\bar{a}(x)=(a_i(x))\in M_f$ with $a_0(x)=x$ . Let $r: M_f\to M_f$ be defined as $r(\bar{y})=\bar{y}^*$ such that $(\bar{y}^*)_i=y_i$ for $i\geq 0$ and $(\bar{y}^*)_i=a_{i}(y_0)$ for $i\leq 0$ . Let

$$u(\bar{y})=\sum_{j=0}^{+\infty}( {\varphi} \circ \sigma_f^j(\bar{y})- {\varphi} \circ \sigma_f^j\circ r(\bar{y}) ).$$

First, we claim that $u: M\to \mathbb{R}$ is well defined and bounded. Indeed, let ${\varphi}$ be Hölder with constant $C_{{\varphi}}>0$ and exponent $\alpha\in(0,1)$ . Then for any $j>0$ , since $(\sigma_f^j(\bar{y}))_i= (\sigma_f^j\circ r(\bar{y}) )_i$ for $i\geq -j$ , we have

\begin{align*} | {\varphi} \circ \sigma_f^j(\bar{y}) - {\varphi} \circ \sigma_f^j\circ r(\bar{y}) | \leq C_{{\varphi}}\cdot \bar{d}^{\alpha}( \sigma_f^j(\bar{y}), \sigma_f^j\circ r(\bar{y}) ) \leq C_{{\varphi}} \cdot \bigg( \sum_{i=j+1}^{+\infty} \frac{1}{2^i} \bigg) ^{\!\alpha} \leq C_{{\varphi}} \cdot 2^{-j\alpha}. \end{align*}

It follows that $u(\bar{y})\leq C_{{\varphi}} \cdot \sum_{j=0}^{+\infty}2^{-j\alpha}<\infty$ .

Then we prove that $r:M_f \to M_f$ is Hölder continuous. Let $\delta_0$ satisfy the size of the Local Product Structure (see Proposition 2.5). Then there exists $C_1=C_1(\delta)>0$ such that, for any ${\varepsilon}<\delta_0$ , any two points $y_0,z_0$ with $d(y_0,z_0)<{\varepsilon}$ and any $\bar{z}\in M_f$ with $(\bar{z})_0=z_0$ , one has that $\mathcal{F}_f^s(y_0,C_1{\varepsilon})\cap \mathcal{F}_f^u(z_0,\bar{z}, C_1{\varepsilon})$ is a single point set. Denote this point by $w_0=w_0(y_0,\bar{z})$ and let the orbit $\bar{w}=(w_i)$ be that given by Proposition 2.5. Let $\lambda :=\textrm{sup}_{\bar{x}\in\mathbb{T}^d_f}\|Df^{-1}|_{E_f^s(\bar{x})}\| $ . Then for $0\leq i\leq i_{{\varepsilon}}:= [\log_{\lambda}({\delta_0}/{2C_1{\varepsilon}})]$ , one has

\begin{align*} d( r(\bar{y})_{-i}, r(\bar{z})_{-i} ) \leq d_{\mathcal{F}_f^s}(r(\bar{y})_{-i}, w_{-i} ) +d_{\mathcal{F}_f^u} (r(\bar{z})_{-i}, w_{-i} ) \leq 2 C_1{\varepsilon}\lambda^i<\delta_0. \end{align*}

So when $\bar{d}(\bar{y},\bar{z})={\varepsilon}\ll\delta_0$ , we have

\begin{align*} \bar{d}(r(\bar{y}), r(\bar{z})) &=\sum_{i=i_{{\varepsilon}}+1}^{+\infty}\frac{d(r(\bar{y})_{-i}, r(\bar{z})_{-i})}{2^i} +\sum_{i=1}^{i_{{\varepsilon}}}\frac{d( r(\bar{y})_{-i}, r(\bar{z})_{-i})}{2^i} +\sum_{i=0}^{+\infty}\frac{d(y_i, z_i)}{2^i},\\ &\leq C\cdot 2^{-i_{{\varepsilon}}} + 2C_1 {\varepsilon} \cdot \sum_{i=1}^{i_{{\varepsilon}}}\frac{\lambda^i}{2^i} + \bar{d}(\bar{y},\bar{z}),\\ &\leq C \cdot \bigg( \frac{2C_1{\varepsilon}}{\delta_0} \bigg)^{\!{\ln 2}/{\ln \lambda} } + 2C{\varepsilon} \bigg( \frac{2C_1{\varepsilon}}{\delta_0} \bigg)^{\!{-}{\ln({\lambda}/{2})}/{\ln\lambda}} +{\varepsilon}, \end{align*}

for some $C>1$ . We can assume that $\lambda>2$ , otherwise the sum $\sum_{i=1}^{i_{{\varepsilon}}}({\lambda^i}/{2^i})$ is controlled by a constant. Then there exist $C_{\delta_0}$ and $0<\beta<1$ such that

(2.23) \begin{align} \bar{d}(r(\bar{y}), r(\bar{z})) \leq C_{\delta_0}\cdot {\varepsilon}^{\beta}= C_{\delta_0}\cdot \bar{d}(\bar{y},\bar{z})^{\beta}. \end{align}

Now we show that u is Hölder continuous. Note that for all $\bar{y}=(y_i), \bar{z}=(z_i) \in M_f$ , one has

(2.24) \begin{align} \bar{d}(\sigma_f(\bar{y}),\sigma_f(\bar{z}) ) \leq 2\bar{d}(\bar{y},\bar{z}). \end{align}

Indeed, this follows from the next two equations

\begin{align*} \sum_{i=1}^{+\infty} \frac{ d( (\sigma_f\bar{y})_{-i}, (\sigma_f\bar{z})_{-i} )}{2^i} = \frac{1}{2}\cdot\sum_{i=1}^{+\infty} \frac{ d( y_{1-i}, z_{1-i} )}{2^{i-1}} =\frac{1}{2}\cdot\sum_{i=0}^{+\infty} \frac{ d( y_{-i}, z_{-i} )}{2^{i}} \end{align*}

and

\begin{align*} \sum_{i=0}^{+\infty} \frac{ d\left( (\sigma_f\bar{y})_i, (\sigma_f\bar{z})_i \right)}{2^i} = 2\cdot\sum_{i=0}^{+\infty} \frac{ d\left( y_{i+1}, z_{i+1} \right)}{2^{i+1}} = 2\cdot\sum_{i=1}^{+\infty} \frac{ d\left( y_{i}, z_{i} \right)}{2^{i}}. \end{align*}

Let ${\varepsilon}:=\bar{d}(\bar{y},\bar{z})$ and ${\varepsilon}_0={\varepsilon}^{1/2}$ . Then for $0\leq j\leq j_0:= [\textrm{ log}_2({{\varepsilon}_0}/{{\varepsilon}})]$ , by (2.24),

(2.25) \begin{align} | {\varphi} \circ \sigma_f^j(\bar{y}) - {\varphi} \circ \sigma_f^j(\bar{z}) | \leq C_{{\varphi}} \cdot (2^j {\varepsilon})^{\alpha} \leq C_{{\varphi}} \cdot {\varepsilon}_0^{\alpha}.\end{align}

Therefore by the choice of $j_0$ and equations (2.23) and (2.25), we have

\begin{align*} |u(\bar{y})- u(\bar{z})| &\leq \sum_{j=0}^{j_0}| {\varphi} \circ \sigma_f^j(\bar{y}) - {\varphi} \circ \sigma_f^j(\bar{z}) | + \sum_{j=0}^{j_0}| {\varphi} \circ \sigma_f^j\circ r (\bar{y}) - {\varphi} \circ \sigma_f^j\circ r (\bar{z}) |\\ &\quad + \sum_{j=j_0+1}^{\infty}| {\varphi} \circ \sigma_f^j(\bar{y})- {\varphi} \circ \sigma_f^j\circ r(\bar{y}) | + \sum_{j=j_0+1}^{\infty}| {\varphi} \circ \sigma_f^j(\bar{z})- {\varphi} \circ \sigma_f^j\circ r(\bar{z}) |,\\ &\leq \sum_{j=0}^{j_0} C_{{\varphi}} \cdot (2^j {\varepsilon})^{\alpha} + \sum_{j=0}^{j_0} C_{{\varphi}} C_{\delta_0} \cdot (2^j {\varepsilon})^{\alpha\beta} + 2\sum_{j=j_0+1}^{\infty} C_{{\varphi}} \cdot 2^{-j\alpha},\\ &\leq C\cdot 2^{j_0\alpha}{\varepsilon}^{\alpha} + C\cdot 2^{j_0\alpha\beta}{\varepsilon}^{\alpha\beta} + C\cdot 2^{-j_0\alpha},\\ &\leq C\cdot {\varepsilon}^{\alpha/2} + C\cdot {\varepsilon}^{\alpha\beta/2} + C\cdot {\varepsilon}^{\alpha/2} \leq C\cdot \bar{d}(\bar{y},\bar{z})^{\alpha_0}, \end{align*}

where $0<\alpha_0<1$ and both $\alpha_0$ and C only depend on ${\varphi}$ and f. Hence u is Hölder continuous.

Finally, let

\begin{align*} \psi(\bar{y})&= {\varphi}(\bar{y})+u\circ \sigma_f(\bar{y})- u(\bar{y}),\\ &= {\varphi}(r(\bar{y}))+ \sum_{j=0}^{+\infty}( {\varphi}(\sigma_f^{j+1}\circ r(\bar{y}))- {\varphi}(\sigma_f^j\circ r(\sigma_f\bar{y})) ). \end{align*}

It follows that $\psi(\bar{y})$ only depends on $\{y_i \}_{i\geq 0}$ and is Hölder continuous.

Proof of Theorem 2.22. By Lemma 2.23, there exist two Hölder functions $u_1$ and $u_2$ defined on $ M_f$ such that $\psi_1={\varphi}_1+u_1\circ\sigma_f-u_1$ and $\psi_2={\varphi}_2+u_2\circ\sigma_f-u_2$ are both independent of negative orbits. This means that $\psi_1$ and $\psi_2$ are actually functions defined on M. Then, by the Livschitz Theorem for Anosov maps, there exists a Hölder function $v: M\to\mathbb{R}$ such that $\psi_1=\psi_2+v\circ f-v$ . Denote the lift of v on the inverse limit space $M_f$ by $v_0$ . It is clear that $v_0: M_f\to \mathbb{R}$ is Hölder continuous under the metric $\bar{d}(\cdot,\cdot)$ . Now, let $u=v_0+u_2-u_1$ , so we have that

$${\varphi}_1={\varphi}_2 +u\circ \sigma_f - u,$$

where $u: M_f\to \mathbb{R}$ is Hölder continuous.

2.4 Topological classification for higher-dimensional Anosov maps

As mentioned in the Introduction, Theorem 1.4 and Corollary 1.5 also hold for d-torus $(d\geq 2)$ on the assumption that $f\in\mathbf{N}^r_1(\mathbb{T}^d)$ $(r>1)$ is irreducible. In this subsection, we state these two results and give the frame to prove them and also Theorem 1.4.

We say that an Anosov map $f\in\mathbf{A}^1(\mathbb{T}^d)$ is irreducible if its linearization $A=f_*: \pi_1(\mathbb{T}^d)\to \pi_1(\mathbb{T}^d)$ has irreducible characteristic polynomial over $\mathbb{Q}$ as a matrix $A\in GL_d(\mathbb{R})\cap M_d(\mathbb{Z})$ .

It is clear that Theorem 1.4 is a direct corollary of the following theorem.

Here we mention that in this d-torus case, we also have leaf conjugacy (see Proposition 2.14) to match the stable periodic data of f and g in the sense of (1.3). The case that g is special has been studied in [Reference An, Gan, Gu and ShiAGGS23] as the following theorem.

• – There exist $C>1, 0<\alpha<1$ such that for any $x\in \mathbb{T}^d$ and $k\in\mathbb{N}$ , its k-preimage set $P_k(x):=\{y\in\mathbb{T}^d\ |\ A^k(y)=x\}$ is becoming dense exponentially, i.e, the set $P_k(x)$ is $C\alpha^k$ -dense in $\mathbb{T}^d$ .

• – The stable foliation $\mathcal{L}^s$ is minimal, i.e., each stable leaf is dense in $\mathbb{T}^d$ .

The first item is a key tool for getting the same stable periodic data from topological conjugacy (see [Reference An, Gan, Gu and ShiAGGS23, Theorem 3.1]) and the second item is useful for the converse (see [Reference An, Gan, Gu and ShiAGGS23, Theorem 5.1]). Note that the proof that the same stable periodic data implies smoothness of the topological conjugacy along stable leaves just needs the minimality of the unstable foliation $\mathcal{L}^u$ , which is independent of the irreducibility (see [Reference An, Gan, Gu and ShiAGGS23, Proposition 5.1]).

The sufficient part in [Reference An, Gan, Gu and ShiAGGS23] relies on the existence of an affine metric on each leaf of the stable foliation $\mathcal{F}_f^s$ , which requires the periodic stable Lyapunov exponents of f actually to be constant. Instead of the above affine metric, we will build a relation between affine structures of $\widetilde{\mathcal{F}}_f^s$ and $\widetilde{\mathcal{F}}_g^s$ via the Livschitz Theorem on the inverse limit space (see Theorem 2.22). For this, we need to match the stable periodic data by the conjugacy on inverse limit spaces, by Corollary 2.20. Hence it suffices to prove the following theorem for obtaining the sufficient part of Theorem 2.24. We mention that the proof in this paper for the sufficient part of Theorem 2.24 is also effective in the case of [Reference An, Gan, Gu and ShiAGGS23].

Theorem 2.27. Let $f,g\in\mathbf{A}^{r}_1(\mathbb{T}^d)\ (r>1)$ be homotopic and irreducible. Let F,G be lifts of f,g respectively by $\pi:\mathbb{R}^d\to \mathbb{T}^d$ . Let $H:\mathbb{R}^d \to \mathbb{R}^d$ be the conjugacy given by Proposition 2.6 such that $H\circ F= G\circ H$ . Denote by $\bar{h}:\mathbb{T}^d_f\to \mathbb{T}^d_g$ the conjugacy induced by H given by Proposition 2.17. If

$$\lambda^s(f,\bar{p})=\lambda^s(g,\bar{h}(\bar{p})), \quad \forall \bar{p}\in \textrm{Per}(\sigma_f), $$

then H is commutative with deck transformations. This implies that f is topologically conjugate to g on $\mathbb{T}^d$ .

We prove this theorem in § 3.

For the necessary part of Theorem 2.24, and especially for the non-special case, we first give a dichotomy for $f\in\mathbf{A}^1_1(\mathbb{T}^d)$ which says that a non-special Anosov map possesses a kind of accessibility.

Przytycki constructed, in [Reference PrzytyckiPrz76], a class of Anosov maps on $\mathbb{T}^d$ such that each of them admits a certain point on which the set of unstable directions contains a curve homeomorphic to an interval in the (dim $M-$ dim $E^s$ )-Grassman space. By this, it is reasonable to use the following definition for the u-accessible class for $f\in\mathbf{A}^1(M)$ .

Definition 2.28. For any $f\in\mathbf{A}^1(M)$ and any $x\in M$ , the u-accessible class of x for f is defined as

\begin{align*} \textrm{Acc}(x,f):=\{ y\in\mathbb{T}^d \; | \; \exists x=y^0, y^1,\ldots,y^k=y \; \; &\textrm{and} \;\; \tilde{y}^i\in M_f \;\; \textrm{with}\;\; (\tilde{y}^i)_0=y^i \\ &\textrm{such that} \;\; y^{i+1}\in\mathcal{F}_f^u (y^i,\tilde{y}^i), \forall 0\leq i\leq k-1 \}. \end{align*}

We say f is u-accessible if $\textrm{Acc}(x,f)=M$ for any $x\in M$ . This means that every pair of points on M can be connected by a path of unstable manifolds with respect to different negative orbits.

We have the following dichotomy which will be proved in § 4.

Let $f,g\in \mathbf{N}^{r}_1(\mathbb{T}^d)\ (r>1)$ be homotopic and irreducible. Since f is special if and only if f is conjugate to its linearization A, by Proposition 2.2, we have that f and g are both special or u-accessible simultaneously, from Proposition 2.29. If f and g are special, then by Theorem 2.25 one has

\begin{align*}\lambda^s_f(p)=\lambda^s(A)=\lambda^s_g(h(p)),\quad \forall p\in\textrm{Per}(f), \end{align*}

where $h\in {\rm Home}_0(\mathbb{T}^d)$ is a conjugacy between f and g. The necessary part of Theorem 2.24 can be deduced from the following theorem.

We mention that in the u-accessible case, there is no need to assume that f is irreducible for the necessary part of Theorem 2.24. However, in the special case, f must be a priori irreducible (see [Reference An, Gan, Gu and ShiAGGS23]).

We also have a version of Corollary 1.5 for the higher-dimensional case. Note that Theorem 2.25 states only the $C^{1+\alpha}$ -regularity of conjugacy along stable leaves. Hence we give the corollary for both the special case and the u-accessible case.

We will give the proofs of Theorem 2.31 and Corollary 2.32 in § 5.

To end this subsection, we explain Theorem 2.24 in the sense of mapping spaces. For a linear Anosov map $A\in\mathbf{N}^1_1(\mathbb{T}^d)\ (d\geq 2)$ , let $f\in\mathbf{A}^r(A)$ and $(f,H)\in\tilde{\mathbf{A}}^r(A)$ be a tuple defined by (1.4), where $H\in\mathcal{H}(f,A)$ with $A\circ H=H\circ F$ for some lift F of f. Say that two tuples $(f_1,H_1), (f_2,H_2)$ are equivalent, which we denote by $(f_1,H_1)\sim_H (f_2,H_2)$ , if $H_2^{-1}\circ H_1: \mathbb{R}^d\to\mathbb{R}^d$ can descend to $h\in\textrm{Home}_0(\mathbb{T}^d)$ . Denote by $\mathbf{T}^r_H(A)$ the space of equivalence classes of such tuples. Denote by $\mathbf{F}^\textrm{H}(\mathbb{T}^d_A)$ the set of Hölder functions on $\mathbb{T}^2_A$ . Define $\tilde{\phi}_1 \sim_{\sigma_A}\tilde{ \phi}_2 $ in $\mathbf{F}^{\rm H}(\mathbb{T}^d_A)$ as

$$\tilde{\phi}_1 \sim_{\sigma_A}\tilde{ \phi}_2 \iff \exists\;\mathrm{H\ddot{o}lder} \;\tilde{u}:\mathbb{T}^d_A\to \mathbb{R}\quad\textrm{such that}\; \tilde{\phi}_1-\tilde{\phi}_2=\tilde{u}\circ \sigma_A-\tilde{u}.$$

Theorem 1.1’ is a direct corollary of the following theorem.

Theorem 2.1’. Let $A\in GL_d(\mathbb{R})\cap M_d(\mathbb{Z})$ induce a non-invertible irreducible Anosov map with one-dimensional stable bundle. Then for every $r>1$ there is a natural injection

\begin{align*} \mathbf{T}^r_{H}(A)\hookrightarrow \mathbf{F}^\textrm{H}(\mathbb{T}^d_A)/_{\sim_{\sigma_A}},\quad\ (f,H)\mapsto \log\|Df|_{E_f^s}\|\circ \bar{h}^{-1}, \end{align*}

where $\bar{h}$ is induced by H via Proposition 2.17.

Proof. Let $f\in\mathbf{A}^r(A)$ , and F,A be lifts of f,A by $\pi$ respectively. Let (f,H) be the tuple such that H is the unique conjugacy between F and A given by Proposition 2.6, such that $A\circ H= H\circ F$ . Let

$$i:(f,H)\mapsto \log\|Df|_{E_f^s}\|\circ \bar{h}^{-1}, $$

where $\bar{h}$ is induced by H via Proposition 2.17.

On the one hand, i naturally induces a well-defined map $\tilde{i}:\mathbf{T}^r_{H}(A)\to \mathbf{F}^\textrm{ H}(\mathbb{T}^d_A)/_{\sim_{\sigma_A}}$ . Indeed, let $(f_1,H_1)\sim (f_2,H_2)$ , and by definition the map $H=H_2^{-1}\circ H_1$ induces a homeomorphism $h\in\textrm{Home}_0(\mathbb{T}^d)$ such that $h \circ f_1=f_2\circ h$ . Then by Theorem 2.31 and the Livschitz Theorem for Anosov maps (see Theorem 2.21), one has that there exists a Hölder function $u:\mathbb{T}^d\to\mathbb{R}$ such that

$$ \log\|Df_2|_{E_{f_2}^s}\|\circ h=\log\|Df_1|_{E_{f_1}^s}\|+u\circ f_1-u. $$

Define $\bar{u}:\mathbb{T}^d_{f_1}\to\mathbb{R}$ as $\bar{u}(\bar{x}):=u( (\bar{x})_0)$ . Then we have

$$\log\|Df_2|_{E_{f_2}^s}\|\circ \bar{h}=\log\|Df_1|_{E_{f_1}^s}\|+\bar{u}\circ \sigma_{f_1}-\bar{u}.$$

Let $\bar{h}_i\ (i=1,2)$ be the conjugacy between $\sigma_{f_i}$ and $\sigma_A$ induced by $H_i$ by Proposition 2.17. Composing $\bar{h}_1^{-1}$ to the last equation, we get

$$\log\|Df_2|_{E_{f_2}^s}\|\circ \bar{h}_2^{-1}=\textrm{ log}\|Df_1|_{E_{f_1}^s}\|\circ\bar{h}_1^{-1}+(\bar{u}\circ\bar{h}_1^{-1})\circ \sigma_A-(\bar{u}\circ\bar{h}_1^{-1}).$$

It is clear that $\bar{u}\circ\bar{h}_1^{-1 }$ is a Hölder function defined on $\mathbb{T}^d_A$ .

On the other hand, $\bar{i}$ is injective. In fact, let $(f_1,H_1)$ and $(f_2,H_2)$ satisfy

$$\textrm{ log}\|Df_1|_{E_{f_1}^s}\|\circ \bar{h}_1^{-1}\sim_{\sigma_A} \log\|Df_2|_{E_{f_2}^s}\|\circ \bar{h}_2^{-1}.$$

Denote $\bar{h}=\bar{h}_2^{-1} \circ \bar{h}_1$ , then one has that

(2.26) \begin{align} \log\|Df_1|_{E_{f_1}^s}\| \sim_{\sigma_{f_1}} \log\|Df_2|_{E_{f_2}^s}\|\circ \bar{h}.\end{align}

Since both $\bar{h}_1$ and $\bar{h}_2$ satisfy Proposition 2.17, the conjugacy $\bar{h}$ can be given by $H_2^{-1}\circ H_1$ by Proposition 2.17. Hence (2.26) actually gives us the condition stated in Theorem 2.27. It follows that $H_2^{-1}\circ H_1$ can descend to $\mathbb{T}^d$ .