Tagged Questions

Question about probability spaces $(X,\mathcal B,\mu)$ with a measurable map $T\colon X\to X$ preserving the measure, that is $\mu(T^{—1}A)=\mu(A)$ for all $A$ measurable.

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Is the average of a dense orbit ergodic for shift function?

Let $\sigma$ be the shift function in the space of two-sided infinite sequences of $\{0,1\}$, $X=\{0,1\}^\mathbb{Z}$ equipped with product topology. We know that there is some point $x\in X$ with ...
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Transitive Actions, Primitive Actions, and Ergodicity

A group action is transitive iff it has one orbit. Intuitively, this seems to say $G$ shuffles around all the elements of the $G$-set. A group action is primitive iff it has no nontrivial blocks, ...
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The measure generated by the Cantor staircase and the intersection of the Cantor set with its translate

Suppose that $T$ is the shift $\bmod 1$ of the Cantor set by an irrational number $\alpha\in (0,1)$. Consider the measure $\mu$ on the interval $[0,1]$ generated by the Cantor staircase. I'd like to ...
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Product of ergodic transformations

I'm asked to give an example, that the product of two ergodic systems is not ergodic in general. I know that for $X_1=X_2=(S^1,B,m,R_a)$ (the irrational rotation on the unit circle with Lebesgue ...
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Entropy of convolution of measures

Let $G$ be a countable, discrete group, and let $\mu_1,\mu_2$ be probability measures on the group $G$. We define the entropy of $\mu_i$ as $H(\mu_i)=\sum\limits_{g \in G}-\mu_i(g)\log(\mu_i(g))$ (...
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Ergodic system has a.e. dense orbits

One more question: Let $X$ be a metric space with probability measure $\mu$ and $T\colon X \to X$ ergodic. $\Rightarrow f$.a.e. $x$ the orbit $O_x=\{T^n(x) : n \in Z\}$ is dense in $X$. So I have ...
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recurrence of a dynamical system on a compact space

I have a question to an exercise which was already posted (but I'm not allowed to comment it). http://mathoverflow.net/questions/145005/poincare-recurrence-theorem-and-convergence-on-compact-metric-...
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Why this two dynamical Systems are not isomorphic?

Given two dynamical Systems on [0,1) with the Borel $\sigma-Algebra$ and the lebegue measure l. $T_a (x) = x + a$ mod1 $T_2 (x) = 2x$ mod1. Show that this two systems are not isomorphic for any ...
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A basic ergodic question

I know that irrational number can be approximated by p/q and error less than 1/q^2. But I still cannot give a rigorous proof to this problem. And how to show that the difference between the left and ...
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Why is the constant here equal to $a$?

Let $(\Omega,R)$ denote a flow, where $\Omega$ is compact metric space and $\gamma$ is a normalized measure on $\Omega$. I have problems to understand the following passage in http://www....
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Ergodic theory for flows: Invariant

Here is the definition of an invariant measure that I do know: Let $(X,\Sigma)$ be a measurable space and let $f\colon X\to X$ be measurable. A measure $\mu$ on $(X,\Sigma)$ is saif to be ...
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How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $M ^ n$ be a Riemannian manifold, $f: M \rightarrow M$ be ...
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What is an unique ergodic measure?

In a book I found the following: [...] on spaces of the form $\Omega\times \mathbb {R}^2$, where $\Omega$ carries a unique ergodic measure. What is meant with $\Omega$ carries a unique ...
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Invariant measure for 1-2x^2 and limit theorem

I have shown that $f(x) = 1-2x^2$ on [-1,1] has an invariant measure equivalent to lebesgue measure via the change of coordinates $h(x) = \sin\pi x/2$. (I.e. $g(x) = h^{-1}( f( h(x)))= 1 -2|x|$ has ...
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Conservative Measures under a group action (reference request)

I was reading a paper and the author define the concept of conservative measure: Let $(X,\mathcal{B})$ a measurable space and $G$ a group that acts on $X$ by $$G\times X:(g,x)\mapsto T_g(x)$$ where ...
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Help with conditional expectation on the circle

Let $p >1$ a integer, $X = \mathbb{R} / \mathbb{Z}$ and $\mu\colon \mathcal{B}\to [0,1]$ a probability measure on the Borel subsets of $X$ which is $T \colon X \ni x \to (px \text{ mod }1)$ ...
Given a stationary and ergodic sequence of real-valued random variables $(X_n)_{n \geq 0}$ and a Borel$(\bar{\mathbb{R}})$-set A, with $\mathbb{P}[X_0 \in A] > 0$, I want to show, that $\mathbb{P}[... 1answer 85 views Transformation preserving the Lebesgue measure Let$f: [0,1] \to [0,1]$a measurable transformation that preserves the Lebesgue measure$\mu$. So we have that$\liminf_n n\mid{f^n(x)-x}\mid\leq1\mu$-almost everywhere. One trying to solve this ... 0answers 20 views Why are weak-mixing systems considered “random” and compact systems considered “ordered”? As I understand it, weak-mixing systems sort of tend to become "orthogonal" to themselves on the long run, and compact systems tend to become almost periodic. How is this related to them being called "... 1answer 53 views Assertions about measures with computers Let's consider the Lebesgue measure ($\mu$) over the closed interval$[0,1]$. As you know,$\mu(\mathbb{Q} \cap [0,1]) = 0$. In other way, as far as I know the computer just can represent accurately ... 0answers 148 views Question about B. Host paper 'Nombres, normaux entropie, translations' I was reading this paper and I got stuck in a detail left for the reader that I couldn't figure out: Let$X = \mathbb{R}/\mathbb{Z}$,$p > 1$a integer,$D_n = \{kp^{-n}\colon 0 \leq k < p^n \}$... 1answer 150 views Uniform Wiener-Wintner Theorem - proof I am looking for proof of uniform version of Wiener-Wintner theorem: Let$(X, \mathcal{A}, \mu, T)$be an ergodic measure preserving system. For$f \in L^1(\mu)$which is orthogonal to the ... 1answer 50 views Operator induced by continuous function and measures If$X$is a compact metric space, and$T:X \rightarrow X$is continuous map, what would be meant by$T_\ast$is the operator on measures induced by$T$? Allow$\mu$to be some Borel regular normed ... 1answer 255 views Not uniquely ergodic transformation Could you teach me an example of NOT uniquely ergodic but ergodic transformation? And when any continuous, measurable, and ergodic transformation on a topological space X is uniquely ergodic, how ... 1answer 95 views Question about Pollicott-Yuri's proof of Rudolph theorem On the book Dynamical System and Ergodic theory by Pollicott and Yuri there is a proof of the Rudolph x2 x3 theorem (page 153). It looks very clean in comparison with the original proof but I didn't ... 2answers 114 views Reference for Algebraic Groups in Ergodic Theory It seems that the theory of algebraic groups is used in ergodic theory. I was hoping someone could recommend an introduction to algebraic groups that assumes a knowledge of commutative algebra ... 1answer 49 views Dynamical Systems Question on Definition I'm working through some old notes on Dynamical systems, and I see a definition that I'm not familiar with. I'll call it property B, because I'm not sure what else to call it. In the notes, we assume ... 1answer 69 views Questions to the proof of the$L^p$Ergodic Theorem of Von Neumann Before giving the Theorem of Von Neumann and asking my questions to its proof, I'll cite the Ergodic theorem of Birkhoff (out of Walters' "An Introduction to Ergodic Theory", p. 34) that is used in ... 1answer 47 views Invariant functions on product of ergodic systems is determined by eigenfunctions? Given an ergodic measure-preserving system$(X,\mathcal{B},\mu,T)$, the product system$(X\times X, T \times T, \mu \times \mu )$need not be ergodic, in other words: It may have non-trivial invariant ... 0answers 32 views Is a discrete random process issued from a sampled continuous ergodic WSS process also ergodic? I have a continuous time process$\{X_t,t\in\mathbb{R}\}$that is WSS and ergodic for the 1st and 2nd moments. I create a random discrete process$\{Y_n,n\in\mathbb{N}\}=\{X_t,t=nT\}$by discretizing ... 0answers 49 views Mixing System and density argument A Mixing system is defined as a dynamical system$(\Omega,\phi^t,\mu)$for which the following relations holds $$1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B);$$ $$2)\qquad\... 1answer 49 views Conditions for “forward” measure-preservation A transformation T being \mu-invariant is by definition a transformation satisfying$$\mu(T^{-1} E) = \mu(E)$$for all measurable sets E. I was wondering what are sufficient conditions for being ... 2answers 85 views Do Anosov flows exist on two dimensional compact manifolds? Question: Let M denote a two-dimensional, smooth, compact Riemannian manifold (without boundary). If \phi:\mathbb{R}\times M \rightarrow M is a smooth flow, then prove that \phi is not an ... 1answer 101 views Ergodic Theorem and flow In Walters' An Introduction to Ergodic Theory on page 34 the Birkhoff Ergodic Theorem is given as follows: Suppose T\colon (X,\mathfrak{B},m)\to (X,\mathfrak{B},m) is measure-preserving (... 0answers 51 views Why is \left\{T^n x|n\geqslant 0\right\} dense in X iff x\in\bigcap_{n\geqslant 1}\bigcup_{k\geqslant 0}T^{-k}U_n? In Walters' An Introduction to Ergodic Theory I found the following Theorem and proof (p. 29): Theorem 1.7. Let X be a compact metric space, \mathfrak{B}(X) the \sigma-algebra of Borel ... 0answers 46 views A basic doubt on Markov chain/ergodicity Consider a finite state (no. of state N) Markov chain \{X_n\} (all the random variables are bounded) such that there is a state i* such that$$ \sum_{i=1}^{N}p_{ii*}^{(n)} > 0$$for all ... 0answers 222 views Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant with ... 1answer 69 views Ergodic (equivalent characterization) Let (\Omega,\mathcal{B},\mu,T) be a measuretheoretical dynamical system. Then this system is called ergodic if$$ B\in\mathcal{B}, T^{-1}(B)=B\implies \mu(B)=0\text{ or }\mu(B^C)=0.$$Show:... 2answers 72 views Numerical integration using Birkhoff theorem There is a method for numerical integration that uses Brikhoff ergodic theorem? For example if we have a irrational number$\alpha$we know that for every continuous function$f \colon [0,1] \to \...
Let $\underline{X}=(X,\mathcal{B},\mu,T)$ and $\underline{Y}=(Y,\mathcal{B},\mu,S)$ be ergodic measure preserving systems on Borel probability spaces. A joining of $\underline{X}$ and $\underline{Y}$ ...