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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10
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605 views

Compact set of probability measures

I think I can solve the following exercise if X is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
6
votes
0answers
133 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 ...
6
votes
0answers
170 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 ...
5
votes
0answers
83 views

Properties of join of open covers

I am studying the basics on Ergodic Theory from Fundamentos da Teoria Ergodica by Oliveira & Viana (you can actually find the pdf on their website), particularly, the properties of topological ...
5
votes
0answers
50 views

Baker's map is ergodic

Define Baker's map by \begin{align} f(x,y) = \begin{cases} (2x,y/2) & \mbox{ if } (x,y)\in[0,1/2]\times[0,1] \\ (2x-1,y/2+1/2) & \mbox{ if } (x,y)\in[1/2,1]\times[0,1] \\ \end{cases} ...
5
votes
0answers
97 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
4
votes
0answers
159 views

Rising Sun Inequality (Dunford-Schwartz maximal inequality)

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be an absolutely integrable function, and let $f^*:\mathbb{R} \rightarrow \mathbb{R}$ be the one-sided signed Hardy-Littlewood maximal function $$f^*(x) := ...
4
votes
0answers
92 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
4
votes
0answers
99 views

Existence of a sequence which is good for mean convergence but not good for pointwise convergence

The ergodic theorems talk about the limit behaviour of the ergodic averages. For example, if $(X,\chi,\mu,T)$ is a measure preserving system, where $\mu$ is a probability measure on $X$, then we have ...
3
votes
0answers
72 views

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 ...
3
votes
0answers
40 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 ...
3
votes
0answers
131 views

Help understanding a geometric proof of the ergodicity of the Gauss measure for continued fractions

Any $x\in(0,1)$ can be written as a (regular) continued fraction $$ x = \cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+\cdots}}} = [a_1,a_2,a_3,\dotsc] $$ An irrational number has a unique expansion, ...
3
votes
0answers
82 views

Strange definition of Ergodicity

In an engineering course, a stationary process was defined to be ergodic if for all $k\in \mathbb{N}$ and for any bounded (measurable) function of $k+1$ variables we have $$\lim_{N\rightarrow ...
3
votes
0answers
48 views

Ergodicity and Appropiate Partition of the Space

I'm trying to solve the following problem. Let $(X, \mathcal{B}, \mu)$ be a probability space and let $T \colon X \to X$ be a measure preserving function. Prove that if $T^n \colon X \to X$ is ...
3
votes
0answers
77 views

Physical interpretation of Ergodicity.

If $R_{\alpha}:[0,1] \to [0,1]$ is defined by $$R_{\alpha}(x)=x+\alpha $$ then $R_{\alpha} $is called a circle rotation, and it is known that $R_{\alpha}$ is ergodic iff $\alpha$ is irrational. I ...
3
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0answers
38 views

Unique ergodicity a spectral invariant?

Suppose $f:X\to X$ and $g:Y\to Y$ are two automorphism of measure spaces. Suppose there is an unitary operator $V:L^2(X)\to L^2(Y)$ satisfying $VU_fV^*=U_g$, here $U_f$ and $U_g$ are the Koopman ...
3
votes
0answers
84 views

Measurability of points regular

I'm reviewing the proof of the theorem of oseledet the book Mañe: Let $M$ a compact metric space and $f:M \rightarrow M$ a homeomorphism, $\pi: F \rightarrow M$ a finite-dimensional continuos vector ...
3
votes
0answers
56 views

Ergodic mean for Schrodinger flow

Let us consider the linear Schrodinger equation in $\mathbb{R}^N$ $$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$ with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ its solution. ...
3
votes
0answers
108 views

Reference for a transformation

Has the (Lebesgue-)ergodic transformation $T: \{0,1\}^{\mathbb{N}} \to \{0,1\}^{\mathbb{N}}$ defined by $T(x(0)x(1)x(2)\cdots) = x(1)x(3)x(5)\cdots$ been well-studied? If so, where? Does it have a ...
3
votes
0answers
123 views

Are geodesic flows on surfaces with negative curvature Anosov?

I'm just going through the original book by Anosov, where he tries to proof this result. I don't quite understand it. So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of ...
3
votes
0answers
145 views

Generalization of Dobrushin's Ergodic Decomposition for continuous Markov Chains

Let $T$ be the shift transformation. Let $P$ be invariant for $T$ and also define a discrete state space Markov Chain. Let $C_{1},\ldots,C_{n}$ be the connected components of the Markov Chain. It ...
3
votes
0answers
133 views

The measures in Furstenberg's correspondence

In the paper Inverting the Furstenberg correspondence (IFC), the author defines a function $D_{A}(\sigma)$ on the Basic clopens of Cantor space, $2^{\mathbb{N}}$, where $A$ is a finite binary string ...
2
votes
0answers
37 views

Over a compact space, the set of continuous functions are everywhere dense in the set of all measurable functions

WARNING: The following post certainly contains too much information. I've included the complete context of my question so that it is clear what I'm talking about and where my question comes from. Feel ...
2
votes
0answers
46 views

Unique Ergodicity

Show that unique ergodicity is a topological invariant. Is arguing as follows an overkill (hopefully if the logic is correct --- I have a feeling that there has to be a way a T-invariant measure has ...
2
votes
0answers
25 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 ...
2
votes
0answers
41 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 ...
2
votes
0answers
114 views

Conditional measure with respect to a sigma-algebra generated by the level sets of a function has full measure on its level set.

Let $(X,\mathscr{B},\mu,T)$ be a measure-preserving system, where $X$ is a compact metric space, $\mathscr{B}$ its Borel $\sigma$-algebra, $\mu$ a Borel probability measure and $T$ continuous. Let ...
2
votes
0answers
44 views

On Ergodicity of the product of two Ergodic transformations

I was recently reading some textbooks and topics in Ergodic theory that I found this fascinating result which I couldn't prove completely: Suppose that $T$ and $S$ are two Ergodic transformations. ...
2
votes
0answers
88 views

Birkhoff Ergodic Theorem Counterexample

I am trying to come up with a counterexample to this theorem under the assumption that the space is not sigma finite. I tried working with the power set of the real numbers with the measure $\mu(A) = ...
2
votes
0answers
47 views

Cesàro Sum of Tangent

Can you proof or disprove the following? $\lim_{n \to \infty} (\frac {\tan1+\tan 2+\cdots+\tan n}{n})=0$. Is there any ergodic type theorem that can come to help?
2
votes
0answers
45 views

Krylov-Bogoliubov theorem without continuity

This question is very closely related to: Continuity in the Krylov-Bogoliubov theorem. The standard counterexample, which is presented in Katok-Hasselblatt is the following: Let $f:[0,1]\rightarrow ...
2
votes
0answers
81 views

non Lebesgue meausre for the Doubling map

Does there exists invariant non Lebesgue probability measures for the doubling map $T:[0,1)\rightarrow [0,1]$ defined by $ T(x)=2x \,\text{mod}(1)? $ So a probability measure different from ...
2
votes
0answers
65 views

Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
2
votes
0answers
76 views

simple proof of the $L^2$ weak law for discrete-time ergodic Markov processes

Let $\{X_t\}_{t\in\mathbb{Z}}$ be a stationary and ergodic stochastic process with finite second moment. Von Neuman's ergodic theorem implies that the time average $(1/N)\sum_{j=0}^{N-1} X_j$ ...
2
votes
0answers
41 views

literature to learn more on ergodic harris recurrent chains with an atom

I'm trying to learn more on the topic mentioned in the title. Namely I'd like to get more information on the behavior of the boundary terms. ie if I decompose sum of my chain (suppose it's ...
2
votes
0answers
76 views

What is the name of this metric: Why is $(\mathcal{M}, L)$ complete

I am reading section 4 of this article about invariant measures: http://maths-people.anu.edu.au/~john/Assets/Research%20Papers/fractals_self-similarity.pdf Let $(X,d)$ a complete metric space, ...
2
votes
0answers
103 views

The chacon transform

I am following this document http://www.jstor.org/stable/2037431?seq=4 Shouldn't it be necessary to check that the chacon transform is ergodic? The theorem I'm familiar is this: Let $T$ be a ...
2
votes
0answers
154 views

Bernoulli shift on $S^\mathbb{Z}$

Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
2
votes
0answers
132 views

Ergodicity and equidistribution

It is known that ergodicity imply dense, but not vice-versa. Also dense does not imply equidistribution (example). But what about equidistribution and ergodicity properties?
2
votes
0answers
82 views

Ergodic/stochastic convergence

I do have a problem with my homework, and to be honest I am simply lacking any idea on how to begin- maybe someone could give me a tip. First off, here is the assignment: The whole assignment deals ...
2
votes
0answers
352 views

Isomorphic measure-preserving systems: circle and torus

According to Definition 2.7 in Ergodic Theory: with a view towards Number Theory, the systems $(X, \mathcal{B}_X, \mu, T)$ and $(Y, \mathcal{B}_Y, \nu, S)$ are isomorphic when there is a $X' \in ...
1
vote
0answers
32 views

Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
1
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0answers
23 views

measure-preserving transformations are spectrally isomorphic

If $(X_{1}, \mathcal{B}_{1}, m_{1})$ and $(X_{2}, \mathcal{B}_{2}, m_{2})$ are probability spaces together with measure-preserving transformations $T_{1}:X_{1}\to X_{1}$,$T_{2}:X_{2}\to X_{2}$. How ...
1
vote
0answers
17 views

Linear affine random dynamical systems - positive Lyapunov index proof check?

Consider the affine random dynamical system $$ X_n = \mathbf A_n X_{n-1} + R_n, $$ starting from an initial non-zero position $X_0$, where $\mathbf A_n\in\mathbb C^{d\times d}$ and $R_n\in\mathbb ...
1
vote
0answers
28 views

T invariant probability measure on a compact space $X$

Let $T:X\to X$ be a continuous map defined on the compact space $X$. I read that if $X$ is a metric space then the set of T-invariant probability measure is not empty. I want to know if this result ...
1
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0answers
36 views

Partitioning a space such that a set of ergodic measures is uniquely supported on one of the sets

I'm reading a book on Ergodic theory, and it says that given a set X with a sigma algebra A, and a measurable automorphism T, then you can take a set of ergodic measures $\mu_i$ $\epsilon$ M$_T(X)$ ...
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0answers
18 views

Question about history of Entropy

I have started to study Ergodic theory and entropy by some books and lecture notes more than three months but unfortunately I'm not familiar with history of Entropy (I know some thing about name of ...
1
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0answers
25 views

Prove a measure is ergodic

Suppose $(X,\mathcal{B})$ be a measurable space, Then assume $T:X\to X$ is uniquely ergodic, i.e. there exists a unique probability invariant measure $m$. Then $m$ is ergodic. Do you have any ideas? ...
1
vote
0answers
34 views

Topologizing ergodic system so that certain function becomes continuous

Let $X$ be a compact metric space and $\mathcal{B}$ its Borel $\sigma$-algebra. Suppose that $(X,\mathcal{B},\mu,T)$ is an invertible ergodic system ($T$ is only a measurable isomorphism, not ...
1
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0answers
26 views

Notation $\mathrm{mod} $ in ergodic theory

Does someone know, what exactly is meant by the following: $$T^{-1}A=A \mod \mu$$ where $\mu$ is a $T$-invariant measure?