Some dynamical properties related to polynomials

2023/04/06 by Wu, Qinqi
#37B05 #37B20 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.03097

Abstract

Let d∈ℤ and p_i be an integral polynomial with p_i(0)=0,1≤ i≤ d. It is shown that if S is thickly syndetic in ℤ, then \(m,n)∈ℤ²:m+p_i(n),m+p₂(n),…,m+p_d(n)∈ S\ is thickly syndetic in ℤ². Meanwhile, we construct a transitive, strong mixing and non-minimal topological dynamical system (X,T), such that the set \x∈ X:∀ open U\ni x,∃ n∈ℤ s.t. Tⁿx∈ U,T²ⁿx∈ U\ is not dense in X.

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