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

TL;DR

This paper investigates the Diophantine properties of interval exchange transformations (IETs), establishing new results on their proximality and connectivity, including optimal shrinking target results for ergodic IETs and contrasting behaviors for 3-IETs.

Contribution

The paper provides new quantitative results on the proximality of ergodic IETs, proves optimality of certain bounds, and shows that no 3-IET is strongly topologically mixing.

Findings

Ergodic IETs satisfy 1 with optimal rate n.

Almost all 3-IETs exhibit divergence in 2 for ps>0.

No 3-IET is strongly topologically mixing.

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 $ \liminf_{n\to\infty}\limits |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.