On uniform recurrence for hyperbolic automorphisms of the $2$-dimensional torus

Abstract

We are interested in studying sets of the form \[ \mathcal{U}(α) := \left\{ x\in X: \ \exists M=M(x) \geq 1 \text{ such that } \forall N\geq M, \ \exists n\leq N \text{ such that } d(T^nx, x) \leq |λ|^{-αN} \right\} \] where $(X,T,d)$ is our metric dynamical system and $|λ|>1$. Although a lot of results exist for the one dimensional case, not as many are known for systems in higher dimensions and especially in the hyperbolic case. We consider $X=\mathbb{T}^2$, $T(x) = Ax \pmod{1}$, where $A$ is a hyperbolic, area preserving, $2\times 2$ matrix with integer entries and $λ$ is the eigenvalue of $A$ of modulus larger than $1$ and we explicitly calculate the Hausdorff dimension of this set.

Figures (5)

• Figure 1: Example for the positions of the fixed blocks for a $1<\theta <1/\alpha$, given that $0\leq \ \alpha \ \leq 3-2\sqrt{2}$ (no overlaps).
• Figure 2: Examples for the positions of the fixed blocks for a $1<\theta <1/\alpha$, given that $\alpha \ \geq 3-2\sqrt{2}$. The first one depicts single overlapping, where the second one depicts double overlapping (see also \ref{['eq. condition for centres, for case 2']} and \ref{['eq. condition about the left ends of consecutive fixed blocks']}).
• Figure 3: A simple depiction of how the fixed blocks in the left-hand side appear: Block A (green) denotes the block we get by the overlapping condition and how it "slides" as we apply the shift operator. The condition $d_{\Sigma}(\sigma^{n_{k}}(\underline{x}), \underline{x})< \lambda^{-\alpha n_{k+1}}$ forces the digits in block A to be equal with the digits in block B. The condition $d_{\Sigma}(\sigma^{n_{k+1}}(\underline{x}), \underline{x})< \lambda^{-\alpha n_{k+2}}$ forces the digits block A to be equal with the digits in block C. Since block A is already predetermined though by block B, block C is a fixed block too.
• Figure 4: Example for the positions of the fixed blocks in the left-hand side, given we have ensured disjointness.
• Figure :

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Proposition 1
• Theorem 3: Snavely
• Corollary 1
• Remark 1
• proof : Proof of \ref{['Equation I: lower bound']}
• Remark 2
• proof : Proof of \ref{['Equation II: lower bound']}
• Remark 3