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Hausdorff dimension of recurrence sets for matrix transformations of tori

Zhangnan Hu, Bing Li

Abstract

Let $T\colon\mathbb{T}^d\to \mathbb{T}^d$, defined by $T x=Ax(\bmod 1)$, where $A$ is a $d\times d$ integer matrix with eigenvalues $1<|λ_1|\le|λ_2|\le\dots\le|λ_d|$. We investigate the Hausdorff dimension of the recurrence set \[R(ψ):=\{x\in\mathbb{T}^d\colon T^nx\in B(x,ψ(n)) {\rm ~for~infinitely~ many~}n\}\] for $α\ge\log|λ_d/λ_1|$, where $ψ$ is a positive decreasing function defined on $\mathbb{N}$ and its lower order at infinity is $α=\liminf\limits_{n\to\infty}\frac{-\log ψ(n)}{n}$. In the case that $A$ is diagonalizable over $\mathbb{Q}$ with integral eigenvalues, we obtain the dimension formula.

Hausdorff dimension of recurrence sets for matrix transformations of tori

Abstract

Let , defined by , where is a integer matrix with eigenvalues . We investigate the Hausdorff dimension of the recurrence set for , where is a positive decreasing function defined on and its lower order at infinity is . In the case that is diagonalizable over with integral eigenvalues, we obtain the dimension formula.
Paper Structure (9 sections, 10 theorems, 112 equations)

This paper contains 9 sections, 10 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Distribution of periodic points
  3. Upper bound on $\dim_{\rm H}R(\psi)$
  4. Lower bound on $\dim_{\rm H}R(\psi)$.
  5. Construct a Cantor set
  6. Construct the mass distribution
  7. Estimation on $\mu(R_{n_j,m})$.
  8. Estimate the local dimension of $\mu$
  9. The proof of Theorem \ref{['theorem2']}

Key Result

Theorem 1.1

Let $(X, \mathscr{B},T, \mu,\rho)$ be a m.m.p.s. Assume that for some $\tau>0$, the $\tau$-dimensional Hausdorff measure $\mathcal{H}^{\tau}$ of $X$ is $\sigma$-finite. Then for $\mu$-a.e. $x\in X$, Futhermore, if $\mathcal{H}^{\tau} (X)=0$, then for $\mu$-almost every $x\in X$,

Theorems & Definitions (21)

  • Theorem 1.1: bos
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 2.1
  • Lemma 2.2: Lemma 2.3 in EW99
  • Lemma 2.3
  • proof
  • ...and 11 more