arXiv subjects
Publications and source records attributed to Jean-Paul Thouvenot.
We prove an analog of Rudolph's theorem for actions of countable amenable groups, which asserts that among invariant measures with entropy at least c on the $G$-shift $(Λ^G,σ)$, a typical measure has entropy $c$ and is Bernoulli. We also address a relative version of this theorem.
We show that every ergodic dynamical system induces a system with pure Lebesgue spectrum of infinite multiplicity.
Answering Vershik's question we show that quasi-similarity does not conserve the entropy, proving quasi-similarity of all Bernoulli actions of a countable infinite group. We prove also the following generalization of Pinsker's theorem: the actions with zero Kirillov-Kushnirenko $P$-entropy and the actions with completely positive $P$-entropy are disjoint. Poisson suspensions are used as examples.
An ergodic dynamical system $\mathbf{X}$ is called dominant if it is isomorphic to a generic extension of itself. It was shown in an earlier paper by Glasner, Thouvenot and Weiss that Bernoulli systems with finite entropy are dominant. In this work we show first that every ergodic system with positive entropy is dominant, and then that if $\mathbf{X}$ has zero entropy then it is not dominant.
We answer positively a question of Ryzhikov, namely we show that being a relatively weakly mixing extension is a comeager property in the Polish group of measure preserving transformations. We study some related classes of ergodic transformations and their interrelations. In the second part of the paper we show that for a fixed ergodic T with property A, a generic extension $\hat{T}$ of T also has the property A. Here A stands for each of the following properties: (i) having the same entropy as T, (ii) Bernoulli, (iii) K, and (iv) loosely Bernoulli.
We study dynamical systems which have bounded complexity with respect to three kinds metrics: the Bowen metric $d_n$, the max-mean metric $\hat{d}_n$ and the mean metric $\bar{d}_n$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_n$ (resp. $\hat{d}_n$) if and only if it is equicontinuous (resp. equicontinuous in the mean). However, we construct minimal systems which have bounded complexity with respect to $\bar{d}_n$ but not equicontinuous in the mean. It turns out that an invariant measure $μ$ on $(X,T)$ has bounded complexity with respect to $d_n$ if and only if $(X,T)$ is $μ$-equicontinuous. Meanwhile, it is shown that $μ$ has bounded complexity with respect to $\hat{d}_n$ if and only if $μ$ has bounded complexity with respect to $\bar{d}_n$ if and only if $(X,T)$ is $μ$-mean equicontinuous if and only if it has discrete spectrum.
We show that for any irrational number $\a$ and a sequence of integers $\{m_l\}_{l\in \N}$ such that $\displaystyle{\lim_{l\to \infty} \norm{m_l \a} = 0}$, there exists a continuous measure $μ$ on the circle such that $\displaystyle{\lim_{l\to \infty} \int_\T \norm{m_l þ} dμ(þ) = 0}$. This implies that any rigidity sequence of any ergodic transformation is a rigidity sequence for some weakly mixing dynamical system. On the other hand, we show that for any $\a \in \R - \Q$, there exists a sequence of integers $\{m_l\}_{l\in \N}$ such that $\norm{m_l \a} \to 0$ and $m_l θ[1]$ is dense on the circle if and only if $þ\notin \Q \a+\Q$.