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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1answer
82 views

Using Kolmogorov's 0-1 law in proof of shift map being ergodic

Why should ${\cal E}_\theta$ be trivial?. I dont see how Kolmogorov's 0-1 law says that in this case we should take the 0 option. This is only mention of ${\cal E}_\theta$ in my notes I can find. ...
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1answer
57 views

For a non-compact metric space, do I have that the set of $\sigma$-invariant measures is compact?

Let $X$ be a non-compact metric space with a sub shift $\sigma: X \to X$. Do I have that the the space of $\sigma$-invariant probability measures on $X$ such that $\mu (B) = \mu (\sigma^{-1}(B))$ with ...
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0answers
68 views

Measure of intermediate local/pointwise dimension is not ergodic for $T_\beta(x)=\beta \cdot x \mod 1$ if $\beta \not \in \mathbb{N}$

Let $\mu$ be an ergodic measure for the map $T_a(x)=a \cdot x \mod 1$ for $a \in \mathbb{N}$ with $a \geq 2$ and invariant for $T_{a^{\frac{1}{n}}}$ for some $n \in \mathbb{N}$ (and thus also ...
2
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1answer
81 views

For a summable function, with summable variation, prove that $\sup_{i \in I} \sup_{x \in [i]}|f(x)| \exp((t-1) \sup_{x \in [i]}f(x) )$ is bounded

$\newcommand{\var}{\operatorname{var}}$ Let $X = \mathbb{N}^\mathbb{N}$ and $f: X \to \mathbb{R}$ be a function such that $$|f|_{\var} = \sum_{i=1}^{\infty} \var_n f < \infty,$$ where $\var_n f = ...
3
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0answers
86 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 ...
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0answers
41 views

Is a pot of boiling water an example of non-ergodic process?

Sorry if this question is a bit dumb... I think (but correct me if I'm wrong) that ice cream moving in a perfect ice cream maker is an example of ergodic flow: the flow itself is conserved, no ...
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0answers
49 views

Is Kolmogorov's zero–one law undecidable?

Kolmogorov's zero–one law is related to other parts of probability like the law of large numbers. However it is stated that what the actual probability of a tail event is (either 0 or 1) is hard to ...
5
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1answer
176 views

Entropy of the induced transformation

I need help with this problem: Let $(X,\mathcal{B},\mu,T)$ be a ergodic dynamical system in the probability space $(X,\mathcal{B},\mu)$. Let $A \in \mathcal{B}$ with $\mu(A)>0$. We define the ...
3
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1answer
64 views

Correlation Sequences and Unitary Operators

Let $U:H \to H$ be an unitary operator on a Hilbert space $H$. Suppose that $x \in H$ is orthogonal to all the eigenvectors of $U$. I'd like to prove that $$ \lim_{N \to \infty} \frac{1}{N} ...
3
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1answer
113 views

Question on Gauss map - application of Birkhoff's ergodic theorem

Take a Gauss map $G: [0,1] \longrightarrow [0,1]$ which is $$G(x) = \frac{1}{x} \mod 1, 0<x<1$$ and $0$ if $x=0$. Let $\mu$ be the Gauss measure. For $x \in [0,1]$ let $[a_{1}(x), ...
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0answers
57 views

Topological entropy, spanning sets and expansiveness of simple maps on a torus

I am trying to solve the following problem. Take the torus $\mathbb{T}^{2}$ and define the map $T(x,y)=(x + \alpha$ mod 1, $x+y$ mod $1)$, where $(x,y) \in [0,1]^{2}$. By induction, we have ...
2
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0answers
116 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 ...
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0answers
74 views

What connections between computer science and ergodic theory?

I have a background in ergodic theory, but I am switching to computer science/programming. I would like to know if tools from ergodic theory could be useful, especially something around coding of ...
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1answer
19 views

$\mathbf{Q}^*$-invariant measurable set of the real line

Let $X$ be some (Lebesgue-)measurable subset of $\mathbf{R}$ such that, for any rational $q \neq 0$, we have $qX=X$. Assume that the Lebesgue measure $\mu(X)$ of $X$ is $>0$. Does it hold then that ...
2
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1answer
94 views

What connections between machine learning and dynamical systems?

I have a background of ("pure") dynamical systems and ergodic theory, but I am switching to machine learning. Can some machine learning questions be treated from a dynamical systems/ergodic theory ...
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1answer
21 views

A problem on Markov process

Suppose, $\Pi_{\theta}$ be the transition probability function of a Markov chain. For any function $f$ define $$\Pi_{\theta}f_{\theta}(x) = \int f(y,\theta)\Pi_{\theta}(x,dy).$$ Is there any ...
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0answers
112 views

Application of Poincare recurrence to Baker's map?

Please see figures at http://www.mpipks-dresden.mpg.de/mpi-doc/kantzgruppe/wiki/projects/Recurrence.html. I heard that one of the applications of the Poincare recurrence theorem (which I do not ...
2
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0answers
47 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. ...
3
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1answer
63 views

necessity of uniform continuity for topological entropy

I am referring to Walters' book "Introduction to Ergodic Theory." When he defines the concept of topological entropy he always assumes that the transformation $T: X \rightarrow X$ is uniformly ...
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1answer
53 views

A question on Bernoulli measures and mixing

this is a question on ergodic theory. Suppose I have an integer $N \geq2$ and a probability space $(\sum^{+} , B, \mu_{p})$, where $\mu_{p}$ is the Bernouilli measure with respect to probability ...
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0answers
90 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) = ...
5
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1answer
104 views

A question on ergodic theory: topological mixing and invariant measures

This is a question on dynamical systems. Suppose I have a compact metric space $X$, with $([0,1], B, \mu)$ a probability space, with $B$ a (Borel) sigma algebra, and $\mu$ the probability measure. ...
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0answers
86 views

Ergodic Theory, Bernoulli Measure, Cylinder Set

Let $N\geq2$ be an integer and consider the probability space ($\Sigma^+$,B, $\mu_p$) where $\mu_p$ is the Bernoulli measure with respect to probability vector $\ p = (p_1,...,p_N)$ . Show that for ...
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0answers
55 views

Commutative Ergodic Theory

I've been working through some of the standard ergodic theorems for class (Birkhoff, Hopf, Chacon-Ornstein), and I'm attempting to extend Hopf's maximal ergodic theorem to the case of having ...
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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 ...
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0answers
33 views

Interpretation of a limit in ergodic theory

I have prove in a exercise that for a ergodic system $(X,\mathcal{B},\mu,T)$ if we take any integrable function $f$ with $\int_X f\,d\mu \neq 0$ then $$\lim_{N\to \infty} \log_N \left\vert ...
3
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0answers
79 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 ...
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51 views

Proving a theorem in Ergodicity

I read the theorem stated below on Wikipedia (http://en.wikipedia.org/wiki/Ergodicity#Formal_definition). But I do not understand how to prove the equivalence of these different definitions.Any hints ...
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1answer
88 views

Markov Chain Ergodic Theorem (Proof references)

Where can I find a proof of the erogidc theorem for Markov chains that doesn't use Birkhoff? The theorem states the following : Let $(X_n)_{n\in \mathbb{N}}$ be an irreducible and positively ...
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2answers
66 views

Connectedness in Measure Preserving Ergodic Systems

In topology a space $X$ is called connected if there is no partition of $X=A\cup B$, that $A$ and $B$ are open. Since ergodic systems have the property that each trajectory visits any neighbourhood of ...
4
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0answers
164 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) := ...
1
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1answer
91 views

Question about Kronecker factor

In his paper Ergodic methods in additive combinatorics, Bryna Kra said that the Kronecker factor $(Z_1, \mathcal{Z}_1, m, T)$ of $(X, \mathcal{X},\mu,T)$ is the sub-$\sigma$-algebra of $X$ spanned by ...
2
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2answers
140 views

Does any measure preserving system have an invertible extension?

Let $\mathsf{X} = \left\{ X,\mathcal{B},\mu,T \right\}$ be any measure preserving system. A sub-$\sigma$-algebra $\mathcal{A}\subseteq \mathcal{B}_X$ with $T^{-1}\mathcal{A}=\mathcal{A}$ modulo $\mu$ ...
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1answer
27 views

How do you get a measure space out of a dynamical system?

I'm reading a book on ergodic theory (by Cesar Silva), and also have read Stein and Shakarchi's third book on undergraduate analysis, where there is a section devoted to some ergodic theory. Both ...
4
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3answers
115 views

Showing $\int^{\frac{\pi}{2}}_{-\frac{\pi}{2}} \log(\cos(\phi))\cos(\phi) \ d\phi = \log(4) - 2 $

This is a minor detail of a proof in 'Chaotic Billiards' by Chernov and Markarian which I foolishly decided to verify. It's page 44 of the book, during the proof that lyapunov exponents exist almost ...
3
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1answer
111 views

Ergodic Rotation of the Torus

Consider the measure preserving dynamical system $(\mathbb{R}^2 / \mathbb{Z}^2, \mathcal{B} \otimes \mathcal{B}, \lambda \otimes \lambda, R_{(\alpha, \beta)})$. This is the torus with the borel ...
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4answers
110 views

Rigour and Formal Reference for Ergodic Theory

I am not even a beginner to Ergodic Theory, but I want to start to read about it. I am coming from a math background and for me its quite important that the definitions to be stated and the formalism ...
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1answer
42 views

Mean Ergodic Theorem for ergodic transformations

I understand how the ergodic averages for an $L^2$ function converge in norm to the orthogonal projection on the space of invariant functions, and I understand how for ergodic transformations this ...
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0answers
37 views

A variation of Birkhoff's Ergodic theorem

Suppose $T_k$ are measure perserving for each $k$, and $f$ is integrable. Is it true that $\frac{1}{n}\sum_{k=1}^n f(T_k\circ T_{k-1}\dots\circ T_1 \omega)$ converge almost surely?
2
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2answers
65 views

What is “abstract” ergodic theory?

This is just a question about the usage of the term "abstract". What kind of questions in ergodic theory is considered "abstract" and what's a "regular" question? From some seminars it seems that ...
3
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2answers
62 views

Measure preserving transform and convergent random variables

I have been trying to learn about Ergodic Theorem for a while and now I have a problem I can't solve. Assume $T$ is a measure preserving transform and $X_n\rightarrow X$ everywhere. Also, assume ...
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0answers
20 views

$L_1$ mean ergodic theorem for the action of compact group

Let $X$ be a Polish space with Borel probability measure $\mu$. A compact group $G$ acts on X continuously. It is right that for any $f\in C_b(X)$ exists a sequence $(g_k \in G)_{k\in \mathbb{N}}$ ...
2
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1answer
114 views

Derivation of the Mexican wave as Mean field equilibrium of an ergodic control problem

I'm dealing with a Mean Field system introduced by Gueant, Lasry, Lions in their "Mean Field Games and Applications" which admits as solution the so-called Mexican wave. Without going into the details ...
4
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1answer
67 views

What keeps measure-preserving transformations from concentrating in a particular portion of a probability space?

I'm trying to show that for an event A with positive probability there is some n bounded by 1/P(A) such that $P(A \cap$ T$^{-n}A) > 0$, where T is a probability-preserving transformation. I'm ...
2
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1answer
62 views

What is the density of the SRB measure conditioned to unstable manifolds?

I have a question regarding the SRB measure. As Lai-Sang Young puts it, the SRB-measure is the invariant ergodic invariant measure most compatible with volume. (see [ ...
2
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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
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1answer
43 views

Question concerning the proof of the Ergodentheorem by Birkhoff

Let $(\Omega,\mathcal{A},\mu,T)$ be an ergodic dynamical system and $f\in L_{\mu}^1$. Then it is a.s. $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f\circ T^k=\int f\, d\mu. $$ Now I ...
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0answers
106 views

Circle rotation (dynamic system)

Here's a passage of my script I do not understand. Define $\Omega:=\left\{z\in\mathbb{C}: \lvert z\rvert =1\right\}$ and consider $a\in [0,1)$. Then $$ T_a:=\Omega\to\Omega, z\mapsto ...
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1answer
44 views

ergodic theorem in the seasonal component analysis of time series

When studying the "seasonal components" part of time series, I once read the following statement. I do not understand what role does the ergodic theorem play here? The decomposition of the process ...
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 ...