On shrinking targets for linear expanding and hyperbolic toral endomorphisms

TL;DR

This paper investigates the Hausdorff dimension of shrinking target sets for linear toral endomorphisms, providing complete results for expanding matrices and partial results with counterexamples for hyperbolic matrices in dimension two.

Contribution

It offers a comprehensive analysis of the Hausdorff dimension for shrinking targets under linear toral endomorphisms, extending previous work and highlighting limitations in higher dimensions.

Findings

Complete dimension results for expanding matrices.

Dimension computation for 2x2 hyperbolic matrices.

Counterexamples to a proposed general dimension formula.

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 $d$.