Mixing and rigidity along asymptotically linearly independent sequences

Abstract

We utilize Gaussian measure preserving systems to prove the existence and genericity of Lebesgue measure preserving transformations $T:[0,1]\to [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically linearly independent sequences. Let ${\lambda }_1,...,{\lambda }_N\in [0,1]$ and let ${\varphi }_1,...,{\varphi }_N:\mathbb{N}\to \mathbb{Z}$ be asymptotically linearly independent (i.e. for any $(a_1,...,a_N)\in {\mathbb{Z}}^N\setminus \{\vec{0}\}$, ${\lim }_{k\to \infty }|{\sum }_{j=1}^N{a}_j{\varphi }_j(k)|=\infty$). Then the class of invertible Lebesgue measure preserving transformations $T:[0,1]\to [0,1]$ for which there exists a sequence $(n_k{)}_{k\in \mathbb{N}}$ in $\mathbb{N}$ with $$\underset{k\to \infty }{\lim }\mu (A\cap T^{-{\varphi }_j(n_k)}B)=(1-{\lambda }_j)\mu (A\cap B)+{\lambda }_j\mu (A)\mu (B),$$ 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. Stëpin (see Theorem 2 in "Spectral properties of generic dynamical systems") and a generalization of a result due to V. Bergelson, S. Kasjan, and M. Lemańczyk (see Corollary F in "Polynomial actions of unitary operators and idempotent ultrafilters").

28 pages