Coexistence of mixing and rigid behaviors in ergodic theory

Abstract

In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary nature. Furthermore, we demonstrate that these characterizations can be employed to obtain various results in the theory of generic Lebesgue-preserving automorphisms of $[0,1]$, IP-ergodic theory, multiple recurrence, additive combinatorics, and spectral theory. As a consequence of one of our results we show that given $(b_1,...b_\ell)\in\mathbb N^\ell$, there is no orthogonal vector $(a_1,\dots,a_\ell)\in\mathbb Z^\ell$ with some $|a_j|=1$ if and only if there is an increasing sequence of natural numbers $(n_k)_{k\in\mathbb N}$ with the property that for each $F\subseteq \{1,...,\ell\}$ there is a $\mu$-preserving transformation $T_F:[0,1]\rightarrow[0,1]$ ($\mu$ denotes the Lebesgue measure) such that for any measurable $A,B\subseteq [0,1]$, $$\lim_{k\rightarrow\infty}\mu(A\cap T_F^{-b_jn_k}B)=\begin{cases} \mu(A\cap B),\,\text{ if }j\in F,\\ \mu(A)\mu(B),\,\text{ if }j\not\in F. \end{cases}$$ We remark that this result has a natural extension to a wide class of families of sequences.