On shrinking targets for linear expanding and hyperbolic toral endomorphisms

Abstract

Let $A$ be an invertible $d\times d$ matrix with integer elements. Then $A$ determines a self-map $T$ of the $d$-dimensional torus $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$. Given a real number $\tau>0$, and a sequence $\{z_n\}$ of points in $\mathbb{T}^d$, let $W_\tau$ be the set of points $x\in\mathbb{T}^d$ such that $T^n(x)\in B(z_n,e^{-n\tau})$ for infinitely many $n\in\mathbb{N}$. The Hausdorff dimension of $W_\tau$ has previously been studied by Hill--Velani and Li--Liao--Velani--Zorin. We provide complete results on the Hausdorff dimension of $W_\tau$ for any expanding matrix. For hyperbolic matrices, we compute the dimension of $W_\tau$ only when $A$ is a $2 \times 2$ matrix. We give counterexamples to a natural candidate for a dimension formula for general dimension $d$.