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Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

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Consider a deterministic map on \mathbb{N} whose behaviour can be projected onto a finite “signature” $$ \sigma(n) = (v_2(n),\ n \bmod 18,\ L_3(n),\ \nu(n)), $$ where $(v_2)$ is the 2-adic valuation, $...
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I'm interested in smooth ergodic theory. Please teach me some recommended books for it. Actually, now I have been reading the supplement of Katok's book, Introduction to the Modern Theory of Dynamical ...
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I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
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Sub-Gaussian concentration for reversible Markov chains with spectral gap Setup. Let $(X_i)_{i\ge1}$ be a stationary, $\pi$-reversible Markov chain on a measurable space with spectral gap $\gamma>0$...
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Setting Let $f\ge 0$ be a measurable function on $[0,\infty)$ and define $g(s,t)=ste^{-s-st}$. For $\lambda>0$ set $$ I(f,\lambda)=\int_0^\infty g(s,f(\lambda s))ds. $$ Let $\mu_T$ be the ...
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Let $(X, \mathcal{B}, \mu)$ be an arbitrary measure space and $T: X \to X$ a measure-preserving transformation. Fix a measurable set $A \in \mathcal{B}$ with $0 < \mu(A) < \infty$, and fix $t \...
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I was reading Ratner's Raghunathan’s topological conjecture and distributions of unipotent flows and confused about a definition in the first page: Ratner used the right action $\Gamma \backslash G$ ...
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I apologize if this is an inappropriately trivial question, but I have a simple question about multiplicative ergodic theorem (MET). I am a non expert trying to learn more about the MET and getting ...
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Consider some ODE given by $$ x'=f(x) $$ for $x\in\mathbf{R}^n$ and $f(x)\in\mathbf{R}^n$ for smooth $f$, and for which all solutions $x(t)$ eventually enter some bounded set. Consider some function $$...
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At the moment I am reading the book "Mathematics of Ramsey Theory" DOI: https://doi.org/10.1007/978-3-642-72905-8 and I am having difficulties with the understanding of proof of the ...
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Recall that for a probability measure $\mu$ on a finitely generated group $G$, the entropy $H(\mu)$ is defined as $$ H(\mu) = - \sum_{ g \in G} \mu(g) \log (\mu(g)). $$ In a paper by Kaimanovich-...
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I've been thinking about structural aspects of conditional expectations. This has resulted in the following surprising approximation result for measurable automorphisms. Let $(X,\Sigma,\mu)$ be a ...
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I'm studying i.i.d. vertex percolation on infinite graphs. Specifically, let $G = (V, E)$ be an infinite connected graph of bounded (finite) degree, where each vertex is independently colored black ...
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I am trying to prove the following conjectured identity: $$ \sup_{n\ge 1, \, 0\le k < 2^n-1} \underbrace{\frac{1}{n}\sum_{j=0}^{n-1} \sin\left( \frac{2\pi k}{2^n-1} 2^j\right)}_{S_n(k)} = \frac{\...
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Let $\Sigma_2 = \{0,1\}^{\mathbb{Z}}$ be the full two-shift with left-shift map $\sigma$ and the standard product metric $$d(x,y) = 2^{-\inf\{|n| : x_n \neq y_n\}}.$$ Fix $\varepsilon = 2^{-m}$ for ...
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For finite-state shift spaces $(X,\sigma)$, we have the variational principle: $$ h_\text{top}(X)=\sup\{h_\mu(\sigma):\mu\text{ is a $\sigma$-invariant ergodic probability measure}\}. $$ From what I ...
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A closely related question was asked by myself and answered by Mikael de la Salle here: Are almost invariant measures for Property (T) groups close to invariant measures? but it turns out I need a ...
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In the collection "Aspects of Positivity in Functional Analysis. Proceedings of the Conference held on the Occasion of H.H. Schaefer's 60th Birthday", there is a contribution by Denes Petz ...
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Let $G$ be a locally compact second countable group with Kazhdan's property (T), acting by homeomorphisms on a compact metrizable space $Z$. We consider the space of Borel probability measures $M_Z$ ...
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While looking at the paper The ergodic theoretical proof of Szemer´edi’s theorem, H. Furstenberg, Y. Katznelson, D. Ornstein, one encounters the following remark (just below Theorem 3.1 in page 533)- ...
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Suppose $(X_n)_{n \in \mathbb{Z}}$ follows a stationary ergodic distribution, where $X_0 \in \{-1,1\}$ has mean $0$. We know that $S_n/n \to 0$ where $S_n = \sum_{i=1}^nX_n$. Can we rule out the ...
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Let $(X, \mathcal{F}, \mu)$ be a $\sigma$-finite infinite measure space, and let $T: X \to X$ be an invertible, measure-preserving transformation. A set $W \subset X$ is called wandering for $T$ if ...
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This is a specific question pertaining to the 'universal' properties of chaos in dynamical systems. Consider a continuous map $T:B\to B$, with $B\subset\mathbb{R}^n$ a compact subset. This defines a ...
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Let $\nu$ and $\mu$ be two probability measures on measurable spaces $(X, \mathcal{A})$ and $(Y, \mathcal{B})$, respectively. A linear operator $$ T : L^1(X, \nu) \to L^1(Y, \mu) $$ is called a Markov ...
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Let $\nu$ and $\mu$ be two probability measures on measurable spaces $(X, \mathcal{A})$ and $(Y, \mathcal{B})$, respectively. A linear operator $$ T : L^1(X, \nu) \to L^1(Y, \mu) $$ is called a Markov ...
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Let $a>0$. Is there $\varepsilon>0$ such that, for all finite groups $G$ and all subsets $A\subseteq G$ with $|A|\geq a|G|$, we have $\frac{1}{|G|}\sum_{g\in G}|A\cap g^2A|\geq\varepsilon|G|$? ...
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Let us call a function $F \colon \mathbb N_{+} \to \mathbb Z$ balanced if for all $n > 1$, the function $\overline{F} = \tau \circ F:\mathbb N_{+} \to \mathbb Z/n$, where $\tau \colon \mathbb Z \to ...
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This question is in the context of Tomita–Takesaki theory. Its brief introduction can be found in wiki and I will borrow terminologies used there. Given $\mathcal{M}$ a $\sigma$-finite factor and $\...
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Consider the following cooperative game played on the circle $S^1$, which we identify with $[0, 1]$ with its endpoints identified. Let $0 < \alpha < 1$ be an irrational number. A total of $N \...
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Let $G$ be a group that acts transitively on $X$. A subset $A\subseteq X$ is thick if any finitely many $G$-translations of $A$ have nonempty intersection. Does the following property have a name? (I ...
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I have been investigating the structure of endomorphisms of compact groups for some time. In particular, I am interested in the relationship between ergodicity and topological transitivity for ...
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Let $T$ be a measure-preserving transformation of a finite measure space $(X,\mathcal F, \mu)$. Then, Poincaré's recurrence Theorem states that, for every positive measurable subset $A\subseteq X$, ...
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In the paper [A] the author proves that ergodic automorphisms of compact groups are isomorphic to Bernoulli shifts. However the proof is discussed only in the abelian case and the last phrase of the ...
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Let $(X,\mu)$ be a probability space with an action $(T_g)_{g\in G}$ of a group $G$ by unitary transformations. Theorem 1.1 in Zorin-Kranich's paper implies that, if $G$ is nilpotent, $H$ is a (...
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Reading some books and papers I found three theorems regarded as "Maximal Ergodic Theorem". I will write the three versions below: Let $X$ be a measure space. Let say that a bounded linear ...
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Let $(X, T, \Sigma, \mu)$ be an ergodic, non atomic measure preserving system with finite measure. Given $f \in L^\infty(X,\mu)$, and $k \in \mathbb Z_+$, we consider the usual pointwise ergodic ...
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The upper Banach density of a set $E\subseteq\mathbb{Z}$ is given by $$d^*(E)=\lim_{N\to\infty}\sup_{M\in\mathbb{Z}}\frac{\#(E\cap\{M+1,\dots,M+N\})}{N}\in[0,1].$$ Is there a set $E\subseteq\mathbb{Z}$...
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Let $(Y,\nu)$ be a $\sigma$-finite infinite measure space and $F$ a measure preserving conservative transformation. If we take a finite measure subset $A, \nu(A)>0$, a Khintchine recurrence theorem ...
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Let $m\ge2$ be an integer and consider the expanding map $E:x\to mx\mod1$ on $[0,1)$. Suppose that the periodic orbits $\{x_j\}_{j=1}^N$ tend to be equidistributed on $[0,1)$ as $N\to\infty$. Then ...
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Let $\mu$ be an invariant probability measure of a hyperbolic toral automorphism (i.e., cat map) $M$ on the two-dimensional torus $\Bbb T^2$. Suppose that the projection of $\mu$ onto the $x$-axis ...
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Given $\Gamma$ a discrete non-amenable group and $\mu$ a probability measure on $\Gamma$, let $\partial\Gamma$ denote the Poisson boundary of the random walk $(\Gamma, \mu)$ and let $\nu$ denote the ...
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In mathematics there have been developed a lot of techniques connected with estimation of various exponential sums. However, I did not succeed in finding the literature(in English) which tells in ...
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Let $(X, \mathcal{F}, \mu)$ be a general measure space, and let $T: X \to X$ be a measure-preserving transformation on $X$. Assume that $T$ is ergodic and satisfies the property that, for any set $A \...
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Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
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In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
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Suppose $(X, \mathcal{M}, \mu, T)$ is a measure-preserving dynamical system with $T$ invertible. I am wondering what properties the dynamical system would need to have in order for the following to be ...
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Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
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Consider a measure space $(X,\mathcal{A},\mu)$ and a measure-preserving transformation $\phi \colon X\rightarrow X$, that is, $\phi$ is measurable and $\phi_*\mu = \mu$. My intuition tells me that we ...
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Where can I find the details of constructing singular continuous ergodic measures for the map $z \mapsto z^2$ on the unit circle? I know that it was done by Furstenberg, but I could not find it ...
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Let $(X,\mathcal{F},\mu)$ be a measure space and let $T:X\to X$ be an ergodic measure-preserving transformation. We assume that $T$ satisfies the property that if $B \in \mathcal{F}$ and $T^{-1}B \...
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