Mixing and rigidity along asymptotically linearly independent sequences

TL;DR

This paper proves that typical measure-preserving transformations on [0,1] can exhibit both mixing and rigidity behaviors along sequences that are asymptotically linearly independent, extending previous results in ergodic theory.

Contribution

It introduces a new class of transformations with combined mixing and rigidity properties along asymptotically linearly independent sequences, generalizing prior work.

Findings

Existence of transformations with specified mixing and rigidity behavior.

Genericity of such transformations in the space of measure-preserving systems.

Refinement and generalization of previous ergodic theory results.

Abstract

We utilize 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 $\lambda_1,...,\lambda_N\in[0,1]$ and let $\phi_1,...,\phi_N:\mathbb N\rightarrow\mathbb Z$ be asymptotically linearly independent (i.e. for any $(a_1,...,a_N)\in\mathbb Z^N\setminus\{\vec 0\}$, $\lim_{k\rightarrow\infty}|\sum_{j=1}^Na_j\phi_j(k)|=\infty$). 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 $$\lim_{k\rightarrow\infty}\mu(A\cap T^{-\phi_j(n_k) }B)= (1-\lambda_j)\mu(A\cap B)+\lambda_j\mu(A)\mu(B),$$ for any measurable $A,B\subseteq [0,1]$ and any $j\in\{1,...,N\}$, is generic. This result is a refinement of a result due to A. M. St\"epin (see Theorem 2 in "Spectral properties of generic dynamical systems") and a generalization of a result due to V. Bergelson, S. Kasjan, and M. Lema\'nczyk (see Corollary F in "Polynomial actions of unitary operators and idempotent ultrafilters").