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      Questions tagged [ergodic-theory]

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

      3
      votes
      0answers
      57 views

      Piecewise linear expanding maps

      Let $(I_{n})$ be a countable infinite disjoint partition of $[0,1)$ into half-open intervals. Let $f:[0,1)\to [0,1)$ be the piecewise linear expanding map with $f(I_{n})=[0,1)$ for all $n$. I suppose ...
      5
      votes
      0answers
      82 views

      $C^{1+\epsilon}$ conjugacy of expanding map on circle

      A continuously differentiable map $f:S^{1}\rightarrow S^{1}$ is called expanding if $|f^{'}(x)|>1$ for all $x\in S^{1}$. We can define the degree of f, def(f) to be number of preimage $f^{-1}(x)$, ...
      1
      vote
      0answers
      61 views

      coboundary in the slow mixing systems

      given dynamical system $(X, T, \mu)$, $\mu$ is probability, $\mu \circ T =\mu$, $T$'s transfer operator $P$ is defined by following relation: $\int (P a) \cdot b d\mu= \int a \cdot (b \circ T) d\mu$ ...
      4
      votes
      2answers
      162 views

      Can one realize this as an ergodic process?

      Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph. We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$ In other words: For ...
      15
      votes
      1answer
      323 views

      Uniform distribution of points on Riemannian manifolds

      Recently, I came across a beautiful paper by Arnol'd and Krylov (Uniform distribution of points on a sphere...) that contains the following theorem: Theorem: Let A and B be two rotations of the ...
      4
      votes
      0answers
      125 views

      mixing time of Young tower

      Given Young tower $(\Delta, m, F)$ with base $\Delta_0$, return map $R$, distortion of $F^R$, g.c.d($R$)$=1$. This system is exact, so mixing, and implies: given fixed $\epsilon$, exists $N$, s.t. $...
      2
      votes
      2answers
      128 views

      Questions about some properties of random probabilities and random expectations

      Let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space with $\mathcal{A}$ countably generated, and let $P: \mathcal{A} \times \Omega \to [0,1]$ be a random probability measure. By that I mean $...
      2
      votes
      2answers
      201 views

      Ergodic theorem and products

      If $f_n (\omega) = \sum_{i=1}^n f_1 (T^i \omega)$ and $T$ is an ergodic action with respect to the measure $\mu$ then it is know as Birkhoff's theorem that $$ \lim_{n \rightarrow \infty} \frac{f_n}{...
      4
      votes
      0answers
      64 views

      Weighted distribution of irrational rotation

      Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, ...
      2
      votes
      0answers
      70 views

      Random Young Tower

      the following setting is the same as Baladi's paper "ALMOST SURE RATES OF MIXING FOR I.I.D. UNIMODAL MAPS" (2002), define random Young tower ($\Delta_{\omega})_{\omega\in \Omega}$ sharing the same ...
      2
      votes
      0answers
      132 views

      Baker map-like problem

      Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...
      5
      votes
      1answer
      158 views

      Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$

      If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to ...
      1
      vote
      2answers
      90 views

      Quantitative bound on irrational rotation recurrence time

      Given an irrational $a$, the sequence $b_n := na$ is dense and equidistributed in $\mathbb S^1$ where we view $\mathbb S^1$ as $[0, 1]$ with its endpoints identified. Given a point $p$ in $\mathbb ...
      5
      votes
      1answer
      95 views

      Connection between rates of convergence in ergodic theorems and spectral gap property

      I've been reading Quantitative ergodic theorems and their number-theoretic applications By Gorodnik and Nevo (arXiv:1304.6847). Early on, there is a comment on rates of convergence in the mean ergodic ...
      1
      vote
      0answers
      92 views

      Ergodic Theorems Birkhoff and Von Neumann

      Is that possible to derive the Birkhoff Ergodic Theorem from or with the help of the Von Neumann ergodic theorem?

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