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arXiv · 2501.11476

On recurrence sets for toral endomorphisms

Zhangnan Hu·Tomas Persson
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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.

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Zhangnan Hu, Tomas Persson. 2025-01-20. On recurrence sets for toral endomorphisms. https://arxiv.org/abs/2501.11476

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