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

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 $\text{lam}$), then the equality $$\begin{array}{cccc}& \underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}n\mathrm{\mid }{T}^n(x)-y\mathrm{\mid }=0& & \text{(A1)}\end{array}$$ holds for $\text{lam}\text{ttimes}\text{lam}$-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 $\text{eps}>0$ $$\begin{array}{cccc}& \underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}{n}^{\text{eps}}\mathrm{\mid }{T}^n(x)-T^n(y)\mathrm{\mid }=\mathrm{\infty },\text{for Lebesgue a. a.}(x,y)\in J^2\mathrm{.}& & \text{(A2)}\end{array}$$ This should be contrasted with the equality $\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\mathrm{\mid }{T}^n(x)-T^n(y)\mathrm{\mid }=0,$ 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.