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Register a new userTopological multiple recurrence of weakly mixing minimal systems for generalized polynomials
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Let $(X, T)$ be a weakly mixing minimal system, $p_1, cdots, p_d$ be integer-valued generalized polynomials and $(p_1,p_2,cdots,p_d)$ be non-degenerate. Then there exists a residual subset $X_0$ of $X$ such that for all $xin X_0$ $${ (T^{p_1(n)}x, cdots, T^{p_d(n)}x): nin mathbb{Z}}$$ is dense in $X^d$.
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In this paper it is shown that every non-periodic ergodic system has two topologically weakly mixing, fully supported models: one is non-minimal but has a dense set of minimal points; and the other one is proximal. Also for independent interests, for a given Kakutani-Rokhlin tower with relatively prime column heights, it is demonstrated how to get a new taller Kakutani-Rokhlin tower with same property, which can be used in Weisss proof of the Jewett-Kriegers theorem and the proofs of our theorems. Applications of the results are given.
The family of pairwise independently determined (PID) systems, i.e. those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages begin{equation*} frac 1 Nsum_{n=0}^{N-1}f_1(T^nx)cdots f_d(T^{dn}x), quad Nto infty, end{equation*} almost surely converge.
Let $(X, mathcal{B},mu,T)$ be an ergodic measure preserving system, $A in mathcal{B}$ and $epsilon>0$. We study the largeness of sets of the form begin{equation*} begin{split} S = left{ ninmathbb{N}colonmu(Acap T^{-f_1(n)}Acap T^{-f_2(n)}Acapldotscap T^{-f_k(n)}A)> mu(A)^{k+1} - epsilon right} end{split} end{equation*} for various families ${f_1,dots,f_k}$ of sequences $f_icolon mathbb{N} to mathbb{N}$. For $k leq 3$ and $f_{i}(n)=i f(n)$, we show that $S$ has positive density if $f(n)=q(p_n)$ where $q in mathbb{Z}[x]$ satisfies $q(1)$ or $q(-1) =0$ and $p_n$ denotes the $n$-th prime; or when $f$ is a certain Hardy field sequence. If $T^q$ is ergodic for some $q in mathbb{N}$, then for all $r in mathbb{Z}$, $S$ is syndetic if $f(n) = qn + r$. For $f_{i}(n)=a_{i}n$, where $a_{i}$ are distinct integers, we show that $S$ can be empty for $kgeq 4$, and for $k = 3$ we found an interesting relation between the largeness of $S$ and the abundance of solutions to certain linear equations in sparse sets of integers. We also provide some partial results when the $f_{i}$ are distinct polynomials.
In this paper we study the topological characteristic factors along cubes of minimal systems. It is shown that up to proximal extensions the pro-nilfactors are the topological characteristic factors along cubes of minimal systems. In particular, for a distal minimal system, the maximal $(d-1)$-step pro-nilfactor is the topological cubic characteristic factor of order $d$.
We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2)$ with two commuting transformations such that for every $0<ell< 4$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0;$$ $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2, T_{3})$ with three commuting transformations such that for every $ell>0$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}Acap T_{3}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0;$$ $bullet$ There exists an ergodic system $(X,mathcal{X},mu,T_1,T_2)$ with two transformations generating a 2-step nilpotent group such that for every $ell>0$, there exists $Ainmathcal{X}$ such that $$mu(Acap T_{1}^{-n}Acap T_{2}^{-n}A)<mu(A)^{ell} text{ for every } n eq 0.$$
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