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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.

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Ergodicity and Poincare's recurrence Theorem

Suppose we have an ergodic measure-preserving dynamical system $T$ on a probability space $(X,\mathcal{F},\mu)$. Let $A\in \mathcal{F}$ with $\mu(A)>0$. Is it possible to find a set $B$ with $\mu(B)...
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Quantum Ergodic Theorem: why $\sqrt{-\Delta}$ is used instead of $-\Delta$?

I'm studying the proof of Quantum Ergodic Theorems in the book Partial Differential Equations II: Qualitative Studies of Linear Equations (3rd edition) by Michael E. Taylor. The book includes the ...
ayphyros's user avatar
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Simple ergodic convergence proof for iid

This is just a simple question from a problem sheet. Consider a sequence of independent identically distributed random variables $Y_0,Y_1,\dots$. Let $f$ be a function such that $\mathbb{E}|f(Y_0)|^2 &...
dlanshiwen's user avatar
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Ergodicity when the variance doesn't converge

A power law (x^(-k)) isn't ergodic for k<= 2 given that the expected value doesn't exist (nothing to converge to). However for 2<k<=3 there is a finite expected value but an infinite variance....
David's user avatar
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2 answers
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Is the translation on torus $\mathbb{T}^2$ ergodic with the haar measure?

Let $f: \mathbb{S}^1 \times \mathbb{S}^1 \to \mathbb{S}^1 \times \mathbb{S}^1$ $f(x,y)=(cx,y)$ such that $c$ is a complex number and $|c|=1$. Consider the haar measure. Is it an ergodic system? Why? I ...
Gabriel Corrêa's user avatar
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Counter-example to $f$ is $T$-invariant iff $f$ is $\mathcal{F}_T$ measurable?

I'm watching a video on The Ergodic Theorem and at around 9:33 he states that $f$ is $T$-invariant iff $f$ is $\mathcal{F}_T$-measurable $T$ is a measure preserving function on $(\Omega, \mathcal{F},...
roundsquare's user avatar
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Do amenable groups have invariant monotiles?

Let $G$ be a countable, discrete, amenable group, and let $B$ a finite subset. We say that $B$ is a monotile if $G$ is a disjoint union of translated copies of $B$. In "Monotilable Amenable ...
Susana Santoyo's user avatar
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Let $\omega =\rho d\theta$ be a volue form of the circle $S^1$. Who are the diffeomorphisms of the circle that let $\omega$ invariant?

Consider the parametrization $\phi:]0,2\pi[\to S^1$ given by $\phi(\theta)=e^{i\theta}$. So the Lebesgue measure is given in local coordinates by the form $d\theta_z(\partial_z)=1$. I know that the ...
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Birkhoff's ergodic theorem holds everywhere

Let $\mu$ be the Lebesgue measure on $(0,1]$. Let $\theta(x) = x + \alpha \mod 1$ for an irrational number $\alpha$. Consider the set $A = (a,b]$ with $0 < a < b < 1$, and write $$S_n(\...
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If $X(t)$ is Ergodic, what about $X^2(t)$?

Given: ${X(t)}$ is W.S.S, Gauss with expected value $=0$, which has $R_{XX}(\tau)$ So $C_{XX}(\tau)=R_{XX}(\tau)$ $\int_0^{\infty}R_{XX}\left(\tau\right)d\tau<\infty$, since $X(t)$ is Ergodic. ...
Analysis_Complex_Study's user avatar
3 votes
1 answer
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When is an ergodic Ito diffusion also mixing

If $(X_t,t\geq0)$ denotes the strong solution to the Ito SDE $\mathrm{d}X_t=b(X_t)\mathrm{d}t+\sigma(X_t)\mathrm{d}W_t$ for $W_t$ the standard $d$-dimensional Brownian motion and $b,\sigma$ satisfying ...
Daan's user avatar
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A trick to prove existence of Haar measure

Let $G$ be an abelian compact and separable group, then there exists a unique Radon measure $\mu$ such that $\mu(g A) = \mu(A)$ for each $g \in G$ and Borel set $A$ $\mu(G) = 1$ The proof of this ...
Paul's user avatar
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Are ergodic continuous time processes (strictly) stationary in the limit?

If $X_t$ is a continuous time Markov process in a general state space $X$, say $X=\mathbb{R}^d$. Is it necessarily true that $X_t$ is stationary in the limit, as I believe that any ergodic process ...
Daan's user avatar
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Application of hyperplane separation theorem and Riesz representation theorem.

Let $X$ be a compact metric space and let $C,D$ be tow disjointed convex compact subsets of $C(X,R)^* $ where the weak $^* -$ topology is used on $C(X,R)^*$,and the interior of C is not empty. As ...
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2 votes
1 answer
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Multiple recurrence property is preserved under invertible extensions

The Multiple Recurrence (MR) Theorem states that for a measure-preserving system $(X, \mathcal{B}, \mu, T)$, and a set $A \in \mathcal{B}$, with $\mu(A) > 0$, and for any $k \in \mathbb{N}$, we ...
antrep1234's user avatar
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Weak ergodicity of discrete birth death process

I am trying to understand the criteria of weak ergodicity through the Dobrushin coeffcient. Let $P(m, k)$ be a transition matrix of a countable state space discrete time Markov chain. Weak ergodicity ...
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Walters Ergodic Theory Theorem 2.4: Is this condition necessary?

Theorem 2.4. of Walters' An Introduction to Ergodic Theory reads as follows: Let $(X_1,\mathbb{B}_1,\mu_1),(X_2,\mathbb{B}_2,\mu_2)$ be probability spaces. Suppose that $V:L^2(m_2)\to L^2(m_1)$ is a ...
Saúl RM's user avatar
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Koopman representation: measurability and continuity, and the unitary interpretation of ergodicity

I am trying to prove the following assertion, stated (but not proved) on pages 13-14 of Bekka, Mayer's Ergodic Theory book (https://doi.org/10.1017/CBO9780511758898): Proposition: Let $G$ be a ...
Chaitanya Tappu's user avatar
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Is the convergence speed of local time in Markov processes a continuum.

By theorem 3.12 (p. 427) in the book Continuous Martingales and Brownian Motion by Revuz and Yor, they claim that for an Harris recurrent Markov process $X$ with invariant measure $\mu$, and two ...
Daan's user avatar
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When is a $\Delta$-skeleton irreducible for diffusion processes?

Say we are given a diffusion process $X$ on $\mathbb{R}^n$, that is positive Harris recurrent. That is, for any measurable set $A$ with $\tau_A=\inf\{t\geq 0:X_t\in A\}$ and some $\sigma$-finite ...
Daan's user avatar
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Positive Harris recurrence and ergodicity for continuous time diffusion processes

For a diffusion process $X=(X_t)_{t\geq0}$ satisfying the usual conditions, we say that $X$ is Harris recurrent if, for a measurable set $A$, the first hitting time $\tau_A=\inf\{t\geq 0:X_t\in A\}$ ...
Daan's user avatar
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Stationarity vs. ergodicity of generalized Kesten process

Not an expert here. I am trying to figure out conditions that guarantee (i) existence of a stationary distribution, and (ii) ergodicity for the following generalized discrete-time Kesten process \...
Harry's user avatar
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Existence of a postive measurable set such that $T^{-k}(E)\cap E=\emptyset$ for a particular $k\ge 1.$

Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left(\{x\in X: T^n(x)=x\}\right)=0$ for every $n\ge 1.$ Let $A\in \...
abcdmath's user avatar
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5 votes
2 answers
127 views

Ergodic series converge to the expectation?

Let $(X_i, Y_i)_{i\in\mathbb{N}}$ be a real-valued stochastic process. We say that $X$ is mean-ergodic, if $$\frac{1}{n}\sum_{i=1}^nX_i\to \mathbb{E}X_1$$ in probability as $n\to\infty$. Let $S_n:=\{i\...
Albert Paradek's user avatar
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Does every $\sigma$-homomorphism of a probability algebra come from a measure preserving transformation on the probability space?

Let $(X,\mathcal{A},\mu)$ be an atomless probability space and let $A\sim B$ whenever $\mu(A\vartriangle B)=0$ for each $A$ and $B$ in $\mathcal{A}$. This way, if $\mathbb{A}$ is the set of $\sim$-...
Susana Santoyo's user avatar
2 votes
0 answers
37 views

Relationship between SRB measures and invariant measures that are absolutely continuous with respect to Lebesgue measure

I am currently studying the LSV map, a variant of the class of Pomeau-Manneville maps, which has the form: $$T(x) =\begin{cases} x+2^\alpha x^{1+\alpha} & 0 \leq x < \frac{1}{2} \\ 2x-1 ...
Ben's user avatar
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Non-uniform averages for ergodic theorem

Theorem (Birkhoff) Let $(E, \mathcal{E}, \mu)$ be a $\sigma$-finite measure space and let $T : E \to E$ be a measure-preserving transformation. Suppose that $f \in L^1(\mu)$. Then, the approximations: ...
legionwhale's user avatar
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Applications, Generalisations and developments of Green-Tao Theorem after 2018

The well-known Green-Tao Theorem is definitely one of the most striking results among different area of Mathematics such as: Number Theory, Combinatorics, Graph Theory, Ergodic Theory,... etc. https://...
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2 votes
1 answer
42 views

Convergence of measures in symbolic dynamics

Let $T$ be the one-sided shift on sequences with $k$ symbols. What are the sufficient conditions for $\mu_n \to \mu$ where $\mu_n$ are all invariant measures? Is it sufficient to show that for each ...
USer12323123's user avatar
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Kingsman subaddtive Ergodic theorem for continuous $f$

Let $X$ be a compact metric space,$T$ a transformation on $X$ and $f_n$ a sequence of continuous subadditive functions $X \to \mathbb{R}$. Des $f_n$ converge pointwise for every $x$? By kingsman ...
Sorfosh's user avatar
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2 votes
1 answer
71 views

Why are rotation numbers not homomorphic?

If $f,g$ are degree-1 monotone maps of the circle, why do we generally have $\rho(f\circ g)\neq\rho(f)+\rho(g)$? I mean, you might say that we have no right to expect an equality. After all, it's not ...
Chris Culter's user avatar
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Projecting onto the Kronecker factor

I'm new here, nice to meet you all. I'm doing a PHD in mathematics, and I need your help. My question is this: Given a measure-preserving system X, And a bounded function f\in\L^\infty(X), the ...
user1304358's user avatar
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Show that $\frac{1}{n}\sum_{j=1}^n T^jx$ converges. $T: \ell^2\to \ell^2$ with $T(x_n) = (a_nx_n)_{n\geq 1}$, $\sup \lvert a_n \rvert \leq 1$

I'm working in "Operator Theoretic aspects of Ergodic theory" by Eisner, Farkas Haase and Nagel. The question is as follows Consider the Hilbert space $H := \ell^2(\mathbb{N})$ and a ...
UpzYaDead's user avatar
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Simplest proof that exactness implies mixing

Let $f$ be a continuous map defined on a compact metric space $X$. Suppose that $f$ preserves the Borel probability measure $\mu$ and that, for every positive-measure set $A\subseteq X$, we have $$\...
Uagi's user avatar
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Why is a rigid rotation on a torus ergodic?

I have the exercise to show that a rigid rotation on a torus is ergodic given that $\langle \omega , j \rangle \notin \mathbb{Z}$ for all $j \in \mathbb{Z}^n$. I tried showing that the for all ...
HalloDu's user avatar
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Probability of All Decimal Digits Appearing in a Ten-Digit-Block of Pi Assuming Normality

In their book A Biography of the World's Most Mysterious Number, Posamentier and Lehmann mentioned that "Conway has indicated that if you separate the decimal value of $\pi$ into groups of ten ...
mathy_mathema's user avatar
2 votes
1 answer
65 views

Every ergodic invariant measure of one dimensional dynamical system is a Dirac measure

Recently, I have learned some theorems about attractors and invariant measures. In the book I am reading, there is a theorem presented without its proof. I am interested in how to prove it. Recall ...
R-CH2OH's user avatar
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1 answer
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For a non-singular transformation $T$ the operator $f\mapsto f\circ T$ is an isometry on $\mathcal L^\infty(X,\mu)$

Let $(X,\mu)$ be a measure space and $T:X\to X$ be a non-singular transformation, that is, $\mu(T^{-1}(A))=0$ if and only if $\mu(A)=0.$ Now consider a map $\Phi:\mathcal L^\infty(X,\mu) \to \mathcal ...
abcdmath's user avatar
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Prove that there exist an $\varepsilon>0$ and $N$, such that for every $n\geq N$, $\mu(A_n\Delta A)\leq \varepsilon$

Lemma 4.7 in Ergodic Theory and Differentiable Dynamics by Racon, makes the statement, Let $A$ be in the $\sigma$-algebra generated by $\lim_{n\to\infty}\bigvee_{i=0}^n\mathscr{A}_i$. Then for all $\...
Nishkarsh's user avatar
1 vote
1 answer
63 views

Examples of specific points $g\in G:=\text{SL}(2,\mathbb R)$ such that the curve $u_tg \Gamma, \Gamma:=\text{SL}(2,\mathbb Z)$ is dense in $G/\Gamma$?

Let $G:=\text{SL}(2,\mathbb R)$, $\Gamma:=\text{SL}(2,\mathbb Z)$ and $u_t=\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}, t\ge 0$ Although this paragraph is not really needed to answer this ...
taylor's user avatar
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1 vote
1 answer
46 views

Properties of converging succession of functionals

Given a sequence of functionals $(\omega_n)_{n \in \mathcal{N}}$ on a Von Neumann algebra $\mathcal{W}$ converging to $\omega$, I have the following doubts: Is it true that $\omega_n$ is pure $\...
MBlrd's user avatar
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Sufficient condition for the convergence of the time average of a dynamical system

Consider a dynamical system defined by a 1-d unimodal map. Suppose that the topological entropy of the system is positive. Then does the time average of f converge to a constant? If so, give me any ...
Tomo's user avatar
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A special case of Birkhoff's ergodic theorem for a nonexpansive map

I have the following difference equation: $x_{t+1}=f(x_t)=x_t+3.61*(1/x_t-1)$ where the domain is the closed interval E=[0.19,17.48]. I know the following facts for this map. This map is smooth and ...
Tomo's user avatar
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2 votes
0 answers
39 views

Application of ergodic theorem to transformed Gaussian process

Assume that $W$ is a zero-mean Gaussian process on $[0,\infty)$ with covariance function satisfying $c(s,t):=E(W(s)W(T))=C(|t-s|)$ for a function $C:[0,\infty) \to \mathbb{R}$. Can I exploit somehow ...
Jack London's user avatar
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6 votes
0 answers
84 views

A maximal invariant generates the invariant $\sigma$-algebra

Let $G$ be a group action on the set $X$, and let $f:(X,\Sigma_X)\to (Y,\Sigma_Y)$ be a measurable maximal invariant ($f$ is constant on each orbit and take different values on different orbits). Am ...
Alphie's user avatar
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When do mixtures of ergodic Markov kernels remain ergodic?

Given two Markov kernels on the same space $\mathfrak X$ and relative to the same dominating measure, $K_0(\cdot,\cdot)$ and $K_1(\cdot,\cdot)$, both ergodic with respective stationary distribution ...
Xi'an ні війні's user avatar
4 votes
0 answers
74 views

Oddity in definition(s) of quasi compact operator

I was wondering about the general definition of a quasicompact operator. There seem to be two main ones floating around in the literature, and I am not sure they are equivalent. The first, and ...
Paul Gullesh's user avatar
3 votes
0 answers
81 views

Uniform law of large numbers for ergodic stationary sequence

I am trying to apply a uniform law of large numbers, which is stated in Lemma 7.2 of "Econometrics" by Fumio Hayashi. The starting point is the stochastic process $\{x_t\}$, which we assume ...
Kristan's user avatar
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31 views

Entropy for flows via reparametrizations

I'm studying the entropy for flows via reparametrizations. Let (X,d) a compact metric space. For a closed interval $I$ which contains the origin, a continuous map $\alpha: I \rightarrow \mathbb{R}$ is ...
felcove's user avatar
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0 votes
1 answer
48 views

Symbolic space and Bernoulli Product measure

Let $\Sigma=\{1,2\}^{\mathbb{N}}$ and $A$ be the sigma-algebra generated by the cylinders sets $\{w∈Ω|∀s∈S,w_s=ϵ_s\}$ with $S⊂\mathbb{N}$ finite and $ϵ_s∈\{0,1\}$ . Let $p∈(0,1)$ . We take product ...
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