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 $(X, \mathcal{X}, μ, T)$ is partially rigid if there is a constant $δ>0$ and sequence $(n_k)_{k \in \mathbb{N}}$ such that $\displaystyle \liminf_{k \to \infty } μ(A \cap T^{n_k}A) \geq δμ(A)$ for every $A \in \mathcal{X}$, and the partial rigidity rate is the largest $δ$ achieved over all sequences. For every integer $d \geq 1$, 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.