On recurrence sets for toral endomorphisms

TL;DR

This work determines the Hausdorff dimension of the recurrence set $R_\tau$ for a toral endomorphism $T$ induced by an invertible hyperbolic $2\times 2$ integer matrix $A$, establishing $\dim_H R_\tau = s_0$ with $s_0 = \min\{ \frac{2\log|\lambda_2|}{\tau+\log|\lambda_2|}, \frac{\log|\lambda_2|}{\tau} \}$, and proving that $R_\tau$ has the large intersection property $R_\tau \in \mathscr{G}^{s_0}(\mathbb{T}^2)$. The proof combines a precise geometric decomposition of recurrence jets $R_n$ into ellipses and comparable parallelograms, a mass-distribution (limsup) approach for the lower bound, and covering arguments for the upper bound, complemented by a uniform distribution modulo $1$ tool. The results extend the authors’ prior work to all invertible hyperbolic $2\times 2$ integer matrices and demonstrate that the same two-term min formula governs the dimension, while also yielding a large intersection class membership. These findings advance the dimension theory for limsup recurrence sets in hyperbolic toral dynamics and provide a framework for related higher-dimensional questions.

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 $τ>0$, let \[ R_τ=\{\, x\in \mathbb{T}^2 : T^nx\in B(x,e^{-nτ})~\mathrm{infinitely ~many}~n\in\mathbb{N} \,\}. \] We calculated the Hausdorff dimension of $R_τ$, and also prove that $R_τ$ has a large intersection property.

Figures (2)

• Figure 1: The ellipse $R_{n,i}$ and parallelograms $E_{n,i}$ and $\tilde{E}_{n,i}$ which are marked in orange for some $n$.
• Figure 2: The ellipse $R_{n,i}$ and parallelograms $\tilde{E}_{n,1}$ and $\tilde{E}_{n,2}$.

Theorems & Definitions (12)

• Theorem 1.1
• Example 1.2
• Remark 1.3
• Lemma 2.2
• proof
• Lemma 2.3: Lemma 3.2 of HPWZ
• Definition 2.4
• Theorem 2.5: Theorem 1.3 in Bugeaud or Example 2.1 in KN
• Lemma 4.1
• proof