Let $d \in \mathbb{N}$, let $A \in GL(d,\mathbb{Z})$ have no eigenvalue of complex modulus $1$, and let $f_A: \mathbb{T}^d \to \mathbb{T}^d$ be the induced hyperbolic toral automorphism defined by
paragraph
admin
\begin{align*}
f_A([y]) = [Ay]
\end{align*}
latex_env
admin
for $[y] \in \mathbb{T}^d$, where $\mathbb{T}^d = \mathbb{R}^d/\mathbb{Z}^d$ and $\pi: \mathbb{R}^d \to \mathbb{T}^d$ denotes the quotient map $\pi(y)=[y]$. Equip $\mathbb{T}^d$ with the quotient metric $d_{\mathbb{T}^d}: \mathbb{T}^d \times \mathbb{T}^d \to [0,\infty)$ defined by
Then there exists $\varepsilon_0>0$ such that for every $\varepsilon \in (0,\varepsilon_0)$ there exists $\delta > 0$ such that every bi-infinite sequence $x: \mathbb{Z} \to \mathbb{T}^d$, written $x_k=x(k)$, satisfying