Multiple partial rigidity rates in low complexity subshifts

Abstract

Partial rigidity is a quantitative notion of recurrence and provides a global obstruction which prevents the system from being strongly mixing. A dynamical system $(X,\mathcal{X},\hat{I}¼,T)$ is partially rigid if there is a constant $\hat{I}´>0$ and sequence $(n_k{)}_{k\in \mathbb{N}}$ such that $\underset{k\to \infty }{\mathrm{liminf}}\hat{I}¼(A\cap T^{n_k}A)\ge \hat{I}´\hat{I}¼(A)$ for every $A\in \mathcal{X}$, and the partial rigidity rate is the largest $\hat{I}´$ achieved over all sequences. For every integer $d\ge 1$, via an explicit construction, we prove the existence of a minimal subshift $(X,S)$ with $d$ ergodic measures having distinct partial rigidity rates. The systems built are $\mathcal{S}$-adic subshifts of finite alphabetic rank that have non-superlinear word complexity and, in particular, have zero entropy.

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