Diophantine properties of IETs and general systems: Quantitative proximality and connectivity

Abstract

We present shrinking targets results for general systems with the emphasis on applications for IETs (interval exchange transformations) $(J,T)$, $J=[0,1)$. In particular, we prove that if an IET $(J,T)$ is ergodic (relative to the Lebesgue measure $\lam$), then the equality \[ \liminf_{n\to\infty}\limits n |T^n(x)-y|=0 \tag{A1} \] holds for $\lam\ttimes\lam$-a. a. $(x,y)\in J^2$. The ergodicity assumption is essential: the result does not extend to all minimal IETs. The factor $n$ in (A1) is optimal (e. g., it cannot be replaced by $n\ln (\ln (\ln n))$. On the other hand, for Lebesgue almost all 3-IETs $(J,T)$ we prove that for all $\eps>0$ \[ \liminf_{n\to\infty}\limits n^\eps |T^n(x)-T^n(y)|= \infty,\quad \text{for Lebesgue a. a.} (x,y)\in J^2. \tag{A2} \] This should be contrasted with the equality $\underset{n\to \infty }{\mathrm{liminf}}|T^n(x)-T^n(y)|=0,$ for a. a. $(x,y)\in J^2$, which holds since $(J^2,T\times T)$ is ergodic (because generic 3-IETs $(J,T)$ are weakly mixing). We also prove that no 3-IET is strongly topologically mixing.

24 pages. Revised version

References in corpus (2)

• A note on zero-one laws in metrical Diophantine approximation
• Recurrence and algorithmic information

Cited by in corpus (3)

• Skew products, quantitative recurrence, shrinking targets and decay of correlations
• Topological mixing for some residual sets of interval exchange transformations
• Bounded type interval exchange maps