Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [ergodic-theory]

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.

1
vote
0answers
22 views

Faithful Representation of von Neumann Algebras

I have questions regarding part of E. Christopher Lance. Ergodic Theorems for Convex Sets and Operator Algebras, Page 205. I am looking for a good reference explaining how the faithful representation ...
1
vote
1answer
28 views

Why is the first return map measure preserving?

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\mc}{\mathcal}$ Definitions and Context Let $(X, \mc X, \mu, T)$ be an invertible measure preserving system and let $A$ be a measurable subset of $X$ with ...
0
votes
1answer
45 views

Is there an invariant measure absolutely continuous wrt to the lebesgue measure for the map f

Let $f:[0,1]\rightarrow[0,1]$ where $f(x)=x/2$ $(1-x)$, and let $\lambda$ be the lebesgue measure on [0,1]. Is there a probability measure $\mu$ that is invariant and absolutely continuous wrt to the ...
2
votes
1answer
75 views

Every quasi-invariant measures is in an invariant measure class

I'm reading "Ergodic Theory and Semisimple Groups" by Zimmer and at the very beginning of Chapter $2$ (pp. $8$) the author claims that An action with quasi-invariant measure can be thought of as ...
0
votes
1answer
33 views

Discrete Ergodic spectrum

$\textbf{Problem}$ An ergodic measure preserving transformation $T$ on $(X,B,\mu)$ is called to have ${discrete \ spectrum}$ if there exists an orthonormal basis for $L^2$ which consists of ...
-2
votes
0answers
30 views

Showing that rigid rotations are distinct. [closed]

Let $T_{\theta}: S^1 = \mathbb{R}/\mathbb{Z} \rightarrow S^1 = \mathbb{R}/\mathbb{Z} \\$ $x \bmod 1 \mapsto x + \theta \bmod 1$. Let $T$ be rigid. Let $\theta$ be irrational. Prove that $0,\ T_{\...
2
votes
0answers
65 views

Birkhoff's Erogdic Theorem and scaling of a random process

I have a question regarding the Birkhoff's Ergodic Theorem and time scaling of a random process: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a given probability space and let $\{ X(k,\omega_k), k\geq 0\}...
0
votes
1answer
29 views

explanation of this example which shows that $(f,\mu)$ ergodic does not imply $(f \times f, \mu \times \mu)$ ergodic

Can anyone help me to understand this example. I have already proved that irrational rotation is ergodic. This example is showing that $f \times f$ is not ergodic. But I am not understanding what this ...
1
vote
0answers
25 views

Reference request: Markov chain

I am in a situation to know whether a discrete time Markov chain evolving on Banach space $\mathbb R^n$ whose evolution (of states) equation we know and then I am interested to project( Hopefully an ...
2
votes
1answer
53 views

Why does this argument for conditional expectation fail?

I have a question and I know it is wrong. However I do not understand where I am messing up. If somebody could explain where I am going wrong, that would be great. If we have a probability space $(X,\...
2
votes
1answer
74 views

What does the statement (Let $K$ be the choquet simplex of all probability measures on $X$) mean?

Let $X$ be a probability measure space. What does the statement (Let $K$ be the choquet simplex of all probability measures on $X$) mean ? It is mentioned in C. Lance. Ergodic Theorems for Convex ...
0
votes
0answers
24 views

What is the expected number of steps that a random walker needs to reach a place with high probability in a 2D lattice?

Let $L_{m,m}$ be a $2D$ lattice. Also, suppose that there is a random walker located in position $(0,0)$. The random walker goes right, left, up, or down randomly in each step and cannot get out of $...
1
vote
0answers
30 views

Is decimal expansion uniquely ergodic?

Let the transformation decimal expansion $f : [0; 1] \rightarrow [0; 1]$, given by $f(x) = 10x − \lfloor 10x\rfloor$ and m the Lebesgue measure in $[0; 1]$. we know that $m$ is invariant by $f$ and ...
2
votes
1answer
31 views

If a set has infinitely many multiples of each integers, then it intersects (S-S) for any set S with positive upper density

I wanted to know whether above statement is true. If it is, how can one go about proving it? Say A $\subset\mathbb{N}$ is a set such that $\forall$ k $\in\mathbb{N}$ , A contains infinitely many ...
0
votes
0answers
10 views

If $μ$ is ergodic then $B(μ)$ has full $μ$-measure [proof verification]

Take $M$ to be a metric space. We call the basin of an invariant probability measure $μ$ the set $B(μ)$ of all points $x ∈ M$ such that $lim _{n→∞} \dfrac{1}{n} \sum_{j=0}^{n−1}ϕ(f ^j(x)) = \int ϕ dμ$...
0
votes
0answers
15 views

show that $f$ is Ergodic with respect of Lebesgue measure.

If $f:[0,1]\to [0,1]$ is given by $$ f(x)= \begin{cases} 2x & \mbox{ if } x\in [0,1/3)\\ & \\ 2x-\frac{2}{3} &...
1
vote
2answers
31 views

If $\chi(g)$ generates a dense subgroup of $\chi(G)$ for all $\chi$ then $g$ generates a dense subgroup of $G$.

This question arised from something in Ergodic theory, however this is not necessary to state or answer the question. Suppose that $G$ is a compact abelian group and $g\in G$. Are the following two ...
0
votes
0answers
19 views

Ergodic Theory: Conditions of the Ergodicity

Let $M$ be a metric space. Prove that an invariant probability measure $\mu$ is ergodic for $f : M \to M \iff$ the time average of every bounded uniformly continuous function $ϕ : M → R$ is constant ...
0
votes
0answers
13 views

The function $φ = \sum _{k=1}^∞ log k \mathcal{X}_{I_k}$ is integrable with relation to Gauss measure?

Prove that for Lebesgue-almost every $x ∈[0,1]$, the geometric mean of the integer numbers $a_1,...,a_n,...$ in the continued fraction expansion of $x$ converges to some real number: in other words, ...
0
votes
1answer
58 views

Weak Convergence of Probability Measures (Proving integrals converge if measures do)

Note: I am doing this question just for fun, not for hw. Question: Fix any dense subset $G$ of the unit ball of $C^{0}(M)$. Here $C^0(M)$ refers to the space of continuous functions defined on $M$, a ...
1
vote
1answer
44 views

Consider the sequence $1,2,4,8,…,a^n = 2^n,…$ of all the powers of $2$ [duplicate]

Consider the sequence $1,2,4,8,...,a^n = 2^n,...$ of all the powers of $2$. Prove that, given any digit $i ∈ {1,...,9}$, there exist infinitely many values of $n$ for which $a^n$ starts with that ...
0
votes
1answer
63 views

The block $617$ occurs infinitely many times in the decimal expansion of almost every $x ∈ [0,1]$

Prove that, for almost everywhere number $x ∈ [0,1]$ whose decimal expansion contains the block $617$ (for instance, $x = 0.3375617264 ···$), that block occurs infinitely many times in the decimal ...
5
votes
1answer
46 views

Ergodic transformation on a atomless measure space

I am currently reading Kakutani–Rokhlin lemma and faced a problem which is given below :--- Let $(X,\mathscr B,\mu,T)$ be an invertible measure preserving system such that $\mu(\{x\})=0,\forall x\in ...
1
vote
0answers
16 views

$\lim_n \sup \frac{1}{n}\# \{0 ≤ j ≤ n − 1 : f^j(x) ∈ D \} > 0$

Let $f:M \to M$ a measurable transformation and $\mu$ a invariant probability and $D \subset M $ a subset of positive measure. Prove that almost every point of $D$ passes a positive fraction of the ...
1
vote
1answer
37 views

What will be a good book to solve exercises from in ergodic theory?

I am now using Walter's book; it has no exercises. I am a graduate student with reasonable background.
4
votes
1answer
36 views

Prove that if $f : M → M$ preserves a probability $\mu$, then for any $k \geq 2 $ $f^k$ preserves $\mu$

Prove that if $f : M → M$ preserves a probability $\mu$, then for any $k \geq 2$, $f^k$ also preserves $\mu$. Is the converse true? Attempt: The first part did induction in $k$. The case $k = 1$ ...
0
votes
0answers
15 views

Proving a method to find the entropy of a measure-preserving function

I am reading chapter 10 this paper and it seems to state but not prove the following theorem. If P generates the σ-algebra of measurable sets, then $h_µ(T ) = h_µ(T,P )$. I understand everything in ...
1
vote
1answer
45 views

If $f$ is ergodic, then is $f_n$ ergodic?

Question: If $(X,E,f,\mu)$ is ergodic ppt, is $(X,E,f_k,μ)$ also ergodic for any $k \in \Bbb N$? I believe this to be true; however, I am having difficulty in proving it. When trying to prove ...
4
votes
1answer
154 views

Convergence of ergodic averages

Let $(X_n)_{n\in\mathbb N_0}$ be a time-homogeneous Markov chain on a probability space $(\Omega,\mathcal A,\operatorname P)$ with transition kernel $\pi$, invariant measure $\mu$ and initial ...
3
votes
0answers
56 views

Reference of the ergodic theorem in continuous time

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\tau_t:\Omega\to\Omega$ for $t\ge0$ such that $\tau_0=\operatorname{id}_\Omega$ $\tau_{s+t}=\tau_s\tau_t$ for all $s,t\ge0$ $\tau:...
1
vote
1answer
40 views

On ergodic theory

Suppose there exist an action of group $G$ on $L^{\infty}(X,\mu)$ via measure preserving transformation ( the left translation Koopmans action). $\mu$ is probability measure. Suppose the action is ...
2
votes
1answer
41 views

Understanding Keane and Petersen's proof of Maximal Ergodic Theorem

The proof is from Easy and nearly simultaneous proofs of the ergodic theorem and maximal ergodic theorem (Keane, Petersen, 2006). Let $(X, \mathcal B, \mu)$ be a probability space, $T:X\to X$ be a ...
0
votes
1answer
90 views

Definition of ergodicity and ergodic process

I am confused by the definitions of ergodicity in wikipedia, see formal definition here which says that a measure-preserving transformation $T$ is ergodic if for every event $E$, $T^{-1}(E) = E$ ...
2
votes
1answer
33 views

Under a measure preserving transformation of a probability space, I need to show the following.

Given a measure preserving system (X,$\mathcal{B}$, $\mu$, T) where $\mu$ is a probability measure, I want to show that $\forall$ A $\in\mathcal{B}$ and $\forall \epsilon >$ 0, $\exists$ n $\in\...
2
votes
1answer
28 views

Koopman-von Neumann Decomposition Properties

Sorry if the title of this question is vague--I'm open to suggestions. For this question, we're working in a probability space $(X, \mathcal{M}, \mu)$. In a proof of "ergodic Roth's theorem" given ...
2
votes
0answers
37 views

Gauss measure and continued fraction

For $x \in [0,1)$ then the continued fraction representation of $$x=0 + \cfrac{1}{a_1(x)+\cfrac{1}{a_2(x)+\cfrac{1}{a_3(x)+\cfrac{1}{\dots}}}}$$ can be written as $[0; a_1(x), a_2(x), a_3(x), \dotsc]$ ...
0
votes
0answers
14 views

Does the property of uniquely ergodic imply that the map has a unique chaotic attractor for all $c$ in $D$

Definition: Let $(X,B)$ be a measurable space and let $T:X→X$ be a measurable transformation. If there is a unique $T$-invariant probability measure then we say that $T$ is uniquely ergodic. Consider ...
3
votes
1answer
68 views

Problem in Ergodic theory

Let $(X,T,\mu)$ be a classical dynamical system, where $(X,\mu)$ is a probability measure space and $T$ is a measure preserving invertible transformation. Let $U$ be the unitary on $L^{2}(X,\mu)$ ...
5
votes
1answer
65 views

Unique ergodicity and first return time

I'm trying to solve the following problem: Let $T\colon X\to X$ be a continuous map on a compact metric space $X$, uniquely ergodic. Let $Y\neq \emptyset$ be an open set. Show that $t(x) = \min\left\{...
0
votes
1answer
33 views

$d(\xi, \eta) = H_{\mu}(\xi|\eta) + H_{\mu}(\eta|\xi)$ defines a metric

I want to show that $d(\xi, \eta) = H_{\mu}(\xi|\eta) + H_{\mu}(\eta|\xi)$ defines a metric on the space of all partitions (considered up to sets of measure zero) of a probability space $(X, \mathscr{...
1
vote
0answers
48 views

About topological conjugacy

The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map? A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ ...
2
votes
0answers
37 views

Von Neumann ergodic theorem for purely continuous spectrum

$\newcommand{\1}{1\negthickspace{\mathrm{I}}}$ Von Neumann ergodic theorem states that, if $U(t)$ is a one-parameter group of unitaries acting on a Hilbert space $\mathcal{H}$, we have $$ \lim_{T\to ...
0
votes
1answer
49 views

Understanding $T(x)=3x\pmod 1$ [closed]

Let $T:[0,1]\to[0,1]$ be such that $T(x)=3x\pmod{1}$ which is measurable with respect to the $\sigma$-algebra of Borel on $[0,1]$, which we denote by $\mathscr{B}_{[0,1]}$. Prove that Lebesgue ...
2
votes
0answers
32 views

Could anyone help me to prove the followings are equivalent?

$K\subseteq \mathbb R^n$ be closed then show that TFAE A probability measure $\mu$ on $K$ is ergodic. (1) Every $\mu$-invariant set of $\mu$ measure zero or one. (2) $\mu$ can-not be decomposed as ...
2
votes
2answers
190 views

What is $T^1(\mathbb H^2/PSL_2(\mathbb Z))$?

Let $\mathbb H^2$ be the upper-half plane. The group $PSL_2(Z)$ acts on $\mathbb H^2$ by isometries, and hence we get an action on $T^1(\mathbb H^2)$. This action is free, smooth, and proper, and thus ...
1
vote
1answer
25 views

Question about proof of equivalent conditions for ergodicity

I am currently attempting to work through Walters' text on Ergodic theory and in the proof of the list of equivalent conditions for ergodicity they state $T^{-n}B\triangle B\subset\cup_{i=0}^{n-1}T^{...
0
votes
0answers
31 views

Is the image of a metric automorphism under a measure-preserving mapping also a metric automorphism?

I'm having my first tiny little bits of ergodic theory so please forgive the probable naivete of the question. So, I'm looking at the first page of chapter 8 of Cornfeld, Fomin, Sinai's "Ergodic ...
1
vote
1answer
52 views

Measure Theory: Simple Measure Invariance Proof

Let M be a metric space, $f: M \rightarrow M$ be a measurable transformation and $\mu$ be a measure on M. Show that $f$ preserves $\mu$ if and only if $\int \phi d\mu =\int \phi \circ f d\mu$ for ...
0
votes
1answer
13 views

Number of preimage in a ergodic map

The problem is to show that if I have $T: (X, \mu) \rightarrow (X, \mu)$ that preserves $\mu$ and is ergodic then the number of preimages of a point is well-defined and is constant $\mu$-ae. I ...
3
votes
0answers
33 views

Definition of ergodic map

I ask the similar question before. About definition of Ergodic theorem. Now just sincerely ask another fundamental problem about the definition of ergodic map. The following definition is what I ...