Topological dynamical systems induced by polynomials and combinatorial consequences

Abstract

Let $d\in N$ and $p_i$ be an integral polynomial with $p_i\left(0\right)=0$, $1\le i\le d$. It is shown that if S is piecewise syndetic in $Z$, then$\left\{\right(m,n)\in Z^2:m+p_1(n),\dots ,m+p_d(n)\in S\}$ is piecewise syndetic in $Z^2$, which extends the result by Glasner and Furstenberg for linear polynomials. Our result is obtained by showing the density of minimal points of a dynamical system of a $Z^2$ action associated with the piecewise syndetic set S and the polynomials $\{p_1,\dots ,p_d\}$.

Moreover, it is proved that if $(X,T)$ is minimal, then for each non-empty open subset U of X, there is $x\in U$ with $\{n\in Z:T^{p_1\left(n\right)}x\in U,\dots ,T^{p_d\left(n\right)}x\in U\}$ piecewise syndetic.