Some dynamical properties related to polynomials

Abstract

Let $d\in \mathbb{Z}$ and $p_i$ be an integral polynomial with $p_i(0)=0,1\le i\le d$. It is shown that if $S$ is thickly syndetic in $\mathbb{Z}$, then $\{(m,n)\in {\mathbb{Z}}^2:m+p_i(n),m+p_2(n),\dots ,m+p_d(n)\in S\}$ is thickly syndetic in ${\mathbb{Z}}^2$. Meanwhile, we construct a transitive, strong mixing and non-minimal topological dynamical system $(X,T)$, such that the set $\{x\in X:\forall \text{open}U\ni x,\exists n\in \mathbb{Z}\text{s.t.}{T}^{n}x\in U,T^{2n}x\in U\}$ is not dense in $X$.

14 pages