arXiv subjects
Publications and source records attributed to Zhangnan Hu.
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.
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.