Mixing and rigidity along asymptotically linearly independent sequences

• Published in: Ergodic theory and dynamical systems Vol. 43; no. 10; pp. 3506 - 3537
• Main Author: ZELADA, RIGOBERTO
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.10.2023
• Subjects: Asymptotic properties; Dynamical systems; Original Article; Polynomials; Prime numbers; Rigidity; Ergodic theory; 37A50; Gaussian systems; generic transformation; rigidity sequence; 37A46; 37A25; 28D05
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2022.71

We use Gaussian measure-preserving systems to prove the existence and genericity of Lebesgue measure-preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically linearly independent sequences. Let $\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _N\in [0,1]$ and let $\phi _1,\ldots ,\phi _N:\mathbb N\rightarrow \mathbb Z$ be asymptotically linearly independent (that is, for any $(a_1,\ldots ,a_N)\in \mathbb Z^N\setminus \{\vec 0\}$ , $\lim _{k\rightarrow \infty }|\sum _{j=1}^Na_j\phi _j(k)|=\infty $ ). Then the class of invertible Lebesgue measure-preserving transformations $T:[0,1]\rightarrow [0,1]$ for which there exists a sequence $(n_k)_{k\in \mathbb {N}}$ in $\mathbb {N}$ with for any measurable $A,B\subseteq [0,1]$ and any $j\in \{1,\ldots ,N\}$ , is generic. This result is a refinement of a result due to Stëpin (Theorem 2 in [Spectral properties of generic dynamical systems. Math. USSR-Izv. 29(1) (1987), 159–192]) and a generalization of a result due to Bergelson, Kasjan, and Lemańczyk (Corollary F in [Polynomial actions of unitary operators and idempotent ultrafilters. Preprint, 2014, arXiv:1401.7869]).