无交换性的多重递推

arXiv - MATH - Dynamical Systems Pub Date : 2024-09-12 DOI:arxiv-2409.07979

Wen Huang, Song Shao, Xiangdong Ye

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引用次数: 0

摘要

本文研究的是无交换性的多重递归。我们证明，对于任意两个同构$T,S：(X,T)$和$(X,S)$都是最小的情况下，存在一个$X$的残余子集$X_0$，对于X_0$中的任意$x\ 和任意非线性积分多项式$p_1,\ldots、p_d$ 在 $0$ 消失时，$\mathbb Z$ 的$\{n_i\}$ 子序列 $n_i\to \infty$ 满足 $$ S^{n_i}x\to x,\ T^{p_1(n_i)}x\to x, \ldots,T^{p_d(n_i)}x\to x,\i\to\infty.$$

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Multiple recurrence without commutativity

We study multiple recurrence without commutativity in this paper. We show that for any two homeomorphisms $T,S: X\rightarrow X$ with $(X,T)$ and $(X,S)$ being minimal, there is a residual subset $X_0$ of $X$ such that for any $x\in X_0$ and any nonlinear integral polynomials $p_1,\ldots, p_d$ vanishing at $0$, there is some subsequence $\{n_i\}$ of $\mathbb Z$ with $n_i\to \infty$ satisfying $$ S^{n_i}x\to x,\ T^{p_1(n_i)}x\to x, \ldots,\ T^{p_d(n_i)}x\to x,\ i\to\infty.$$

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arXiv - MATH - Dynamical Systems

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