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 $T:[0,1]\rightarrow [0,1]$ 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").