Hausdorff dimension of recurrence sets for matrix transformations of tori

2024/02/07 by Zhangnan Hu, Bing Li, Hu, Zhangnan +1
Mathematics · #28A80 #37B20 #37C45 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.04810

openalex publication_date 2024/02/07 · openalex created_date 2024/02/09 · openalex updated_date 2026/07/28

Abstract

Let T\colon\mathbbT^d→ \mathbbT^d, defined by T x=Ax(\bmod 1), where A is a d× d integer matrix with eigenvalues 1<|λ₁|≤|λ₂|≤…≤|λ_d|. We investigate the Hausdorff dimension of the recurrence set R(ψ):=\x∈\mathbbT^d\colon Tⁿx∈ B(x,ψ(n)) \rm ~for~infinitely~ many~n\ for α≥log|λ_d/λ₁|, where ψ is a positive decreasing function defined on ℕ and its lower order at infinity is α=\liminf_n→∞(-log ψ(n))/(n). In the case that A is diagonalizable over ℚ with integral eigenvalues, we obtain the dimension formula.

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