On recurrence sets for toral endomorphisms

TL;DR

This paper investigates the recurrence properties of a specific toral endomorphism, calculating the Hausdorff dimension of certain recurrence sets and establishing their large intersection properties.

Contribution

It provides explicit Hausdorff dimension calculations and large intersection results for recurrence sets under toral endomorphisms with eigenvalues less than one.

Findings

Calculated Hausdorff dimension of recurrence sets

Proved recurrence sets have large intersection properties

Analyzed recurrence behavior for specific toral endomorphisms

Abstract

Let $A$ be a $2\times 2$ integral matrix with an eigenvalue of modulus strictly less than 1. Let $T$ be the natural endomorphism on the torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$, induced by $A$. Given $\tau>0$, let \[ R_\tau =\{\, x\in \mathbb{T}^2 : T^nx\in B(x,e^{-n\tau})~\mathrm{infinitely ~many}~n\in\mathbb{N} \,\}. \] We calculated the Hausdorff dimension of $R_\tau$, and also prove that $R_\tau$ has a large intersection property.