I've been trying to find in the literature a rigorous formal proof for the next two equivalences:
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$d$-dimensional toral automorphisms are expansive iff they are hyperbolic.
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$d$-dimensional toral endomorphisms are forward expansive iff all of its eigenvalues have absolute value greater than 1.
The definitions are:
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$T$ is hyperbolic if $T(x) = Ax \bmod 1$ for $A \in \operatorname{SL}(d,\mathbb{Z})$ and $A$ has no eigenvalue with unit absolute value.
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$T$ is expansive if $\exists \epsilon\, \forall x\neq y \in X\, \exists n \in \mathbb{Z}: d(T^n , T^n y) > \epsilon$.
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$T$ is called forward expansive whenever: the same as for expansive just that the index $n$ is in $\mathbb{N}$ instead of $\mathbb{Z}$.
I am looking for a good reference for the proof of these equivalences.
Best Answer
I am not an expert in this area, but I happen to see some references recently for equivalence 1. I am not sure about equivalence 2. The following are references from the texts. My comments are given as remarks.
References from [1]
Let $\Bbb T^d=\Bbb R^d/\Bbb Z^d$ denote the additive $d$-dimensional torus. This is a compact abelian group, and any continuous surjective endomorphism $T:\Bbb T^d\to \Bbb T^d$ is of the form $$T(x)=T_A(x),$$ where $A$ is a non-singular $d\times d$ integer matrix, and $T_A$ is defined by $$T_A \begin{bmatrix}x_1\\\vdots\\ x_d\end{bmatrix}=A\begin{bmatrix}x_1\\ \vdots\\ x_d\end{bmatrix} \pmod 1$$
Assume the transformation $T_A$ is an automorphism (that is, $\det(A)$ is $\pm 1$.) Then the transformation $T_A$ is said to be $\textit{expansive}$ if there is a constant $\delta>0$ with the property that for any pair $x\neq y\in \Bbb T^d$, there is an $n\in\Bbb Z$ with the property that $$\rho(T_A^n x,T_A^n y)>\delta$$
$\textbf{Lemma 2.9}$ The invertible transformation $T_A$ is expansive if and only if no eigenvalue of $A$ has unit modulus.
This well-known result may be seen directly using the Jordan form of the complexified matrix (outlined in [2], chapter 8) or as part of the general theory of hyperbolic dynamical systems (see [3]). Eisenberg in [4] proved the analogous statement for matrices acting on topological vector spaces (which for real vector spaces implies the toral result) over any non-discrete topological field.
Remark: Here the conditions $\det(A)=\pm 1$ and eigenvalues of $A$ are not unit modulus will correspond to your stated definition of hyperbolic. The text only gives proof for dimension 2, although it does provide references for the general case. Also, you can download [4] freely.
References from [2] (page 143)
Let $A$ be an automorphism of the $n$-torus, and $[A]$ the corresponding matrix. Then $[A]$ is expansive iff $[A]$ has no eigenvalues of modulus 1.
Proof:
Let $d$ be the metric given by $$d(\{x_n\},\{y_n\})=\sum_{n=-\infty}^\infty \frac{|x_n-y_n|}{2^{|n|}}.$$ Suppose $\{x_n\}\neq \{y_n\}$. Then for some $n_0,x_{n_0}\neq y_{n_0}$ and \begin{align*} d(T^{n_0}\{x_n\},T^{n_0}\{y_n\}) &=\sum_{n=\infty}^\infty \frac{1}{2^{|n|}} |x_{n+n_0} - y_{n+n_0}|\\&\geq |x_{n_0}-y_{n_0}|\\&\geq 1.\end{align*} Thus 1 is an expansive constant.
Remark: This seems to answer equivalence 1 exactly, where you set $\epsilon = 1$.
Remark While [3] also appears to contain all the information for equivalence 1, I was unable to pull out a single proposition/theorem proving it as the hyperbolic toral automorphisms are considered as examples there.
References
[1] G. Everest and T. Ward, Heights of Polynomials and Entropy in Algebraic Dynamics, Universitext, Springer, 1999
[2] P. Walters, An Introduction to Ergodic Theory, Springer, New York 1982.
[3] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995.
[4] M. Eisenberg, Expansive automorphisms of finite-dimensional spaces, Fundamenta Math. LIX (1966), 307-312