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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11 views

An ergodic stationary stochastic process

Can you give me an example of discrete-time stochastic process on $\mathbb{R}_+$ that is stationary and ergodic but does not have independence in time?
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Stationary distribution in a Markov process.

Consider the homogeneous Markov process with matrix $W$ which describes the "probability of transition" to pass from a state $a$ to $b$. So in the time $t+1$ the probability to be in the state $a$, $...
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For ergodic Markov chains, when does $\lim_{N\to\infty} \mathbb{E}[\sum_{n=1}^{N}f(X_n)] - N\mu(f)$ exist

For an ergodic Markov chain (process would be even better) $X_{n}$ with stationary distribution $\mu$, under which conditions does $$ L:=\lim_{N\to\infty} \mathbb{E}[\sum_{n=1}^{N}f(X_n)] - N\mu(f) $$ ...
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Show that Ergodic Theorem is a special case of Kingman's Subadditive Ergodic Theorem.

A version of Birkhoff's Ergodic Theorem is the following: Theorem 1: Take $\xi\in L^{1}(\Omega,\mathcal{F},\mathbb{P})$. If $\theta$ preserves $\mathbb{P}$, then $$\dfrac{\xi(\omega)+\xi(\theta\...
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23 views

Can we deduce that $\mu(X)\leq \mu(Y)$?

Given a measurable function $f: X \to Y$, if $f$ is injective and measure-preserving that $\mu(f(A))=\mu(A)$ for all subsets $A$ and $\mu$ is a probability measure, can we deduce that $\mu(X)\leq \mu(...
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Set of weakly mixing transformations is not empty

I'm currently trying to understand Halmos' proof of the fact, that the set of weakly mixing Transformations is a dense $G_\delta$ set in the set of measure-preserving transformations. In his proof (in ...
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Proving that the basin of an invariant probability measure has full measure

I'm trying to solve problem 4.1.5 from the book Foundations on ergodic theory from M. Viana and K. Oliveira. The problem is stated as follows: Let $M$ be a metric space. We call the basin of an ...
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Hilbert space version of the notion of conditionally weakly-mixing functions.

$\newcommand{\norm}[1]{\|#1\|}$ $\newcommand{\ab}[1]{\langle #1\rangle}$ $\newcommand{\mr}{\mathscr}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\E}{\mathbb E}$ $\newcommand{\C}{\mathbf C}$ ...
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23 views

Birkhoff ergodic theorem on a lattice

Let $\mathbb{P}_0$ be a probability measure on $\mathbb{R}$. Let $\Omega = \mathbb{R}^{\mathbb{Z}^d}$ and $\mathbb{P} = (\mathbb{P}_0)^{\otimes \mathbb{Z}^d}$ so the the canonical process $X:\Omega \...
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Time averages for a 2-dimensional harmonic oscillator

I'm studying Ergodic Theory and I think I "got" the concept, but I need an example to verify it. Let's take the simplest possible 2D classical harmonic oscillator whose kinetic energy is $$T=\frac{\...
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Relation between $\sigma$-algebras generated by $L^2$-eigenfunctions and $L^\infty$-eigenfunctions.

$\newcommand{\mc}{\mathcal}$ $\newcommand{\C}{\mathbb C}$ Let $(X, \mc X, \mu)$ be a probability space and $T:X\to X$ be an invertible measure preserving transformation. Let $U_T$ be the associated ...
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Difference kind of entropy for Zd actions

I am studying entropy for $\mathbb{Z}^d$ e $\mathbb{N}^d$. I learned that there are problems in using the standard definition. Those problems are related to the smooth representation problem and the ...
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89 views

Spectral description of the Kronecker factor

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\Z}{\mathbb Z}$ $\newcommand{\C}{\mathbb C}$ $\newcommand{\R}{\mathbb R}$ Definitions Let $(X, \mc X, \mu)$ be probability ...
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34 views

On the Equidistribution Weyl Theorem for $\{2^kx\}$

It is well known that the sequence $\{2^k \eta\} \bmod 1$ is uniformly distributed for almost all, but not all, irrational $\eta$ in $(0,1)$. If I fix an irrational number $0<x<1$ ($x$ is ...
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53 views

$\mathcal{M}(X)$ compact in Weak* Topology

If $\mathcal{M}(X)$ is the space of all Borel probability measures on $X$ with Weak$^{∗}$ topology. Is there an example where $\mathcal{M}(X)$ is compact, but $X$ is not metric or not compact?
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Induced $σ$-algebras of random process

Let $\xi_t$, $t\in\mathbb{R}$ be a random process, and $\mathcal{F}_{=t}$, $\mathcal{F}_{≥t}$,... be the induced $σ$-algebras. Check equivalence of the following properties: 1) $∀t ∈ \mathbb{R}$, ...
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References for slow entropy and netropy for Nd and Zd actions

I am studying entropy. Could you give me some references for the the generalizations of entropies, both measure-theoretical and topological, for $\mathbb{Z}^d$ and $\mathbb{N}^d$ actions and for the ...
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If a metric space $M$ is not compact then the space of probability measures $\mathcal M_1(M)$ is not compact.

Let $M$ be a metric space. Then we define the space $\mathcal M_1(M)$ as the topological space $$\mathcal M_1(M) :=\left\{\mu;\ \mu\ \text{is a } \text{Borel probability measure on }M\right\} $$ ...
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Combining Markov chain transitions

Suppose I have two Markov chains with transition densities $\mathbb{Q}(x'\vert x)$ and $\tilde{\mathbb{Q}}(x'\vert x)$. Let's suppose that the Markov chain corresponding to $\mathbb{Q}$ satisfies ...
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24 views

atomic ergodic measure

Let $m$ be a atomic Borel probability measure on $X$. Let $\phi$ be a homeomorphism on $X$. I want to prove that $m$ is $\phi$ Ergodic if and only if $m$ is concentrated on a single $\phi$ orbit. I ...
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256 views

Example of a compact topological space $M$ such that $\mathcal M_1(M)$ is not compact.

It is well known that if $(M,\tau)$ is a compact Hausdorff topological space then (by Riesz–Markov–Kakutani representation theorem + Banach–Alaoglu theorem) we have that the space $$\mathcal M_1(M) :=...
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55 views

Show ergodicity of $2x\operatorname{mod}1$

Let $(E,\mathcal E,\mu)$ denote the Lebesgue measure space on $[0,1)$, $$\tau(x):=2x-\lfloor 2x\rfloor\;\;\;\text{for }x\in E,$$ $$Y_0:=\lfloor 2x\rfloor\;\;\;\text{for }x\in E$$ and $$Y_n:=Y_0\circ\...
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21 views

Ergodic measure

Let $m$ be a $\phi$ Ergodic measure on a space $X$, where $\phi$ is a homeomorphism on $X$. Let $\mu_1$, $\mu_2$ be two $\phi$-invariant $\sigma$ finite measures on $X$ such that $m$ is equivalent to ...
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29 views

What are co-boundaries in analysis?

(Context: my lecturer's notes on Von Neumann's mean ergodic theorem) Given a measure-preserving map $T$ on $(X,\mathcal{A},\mu)$ and an $L^2(X,\mathcal{A},\mu)$ function $f$, we define the linear ...
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Is there a von Neumann type theorem for the $\sigma$-algebra generated by the set of all the eigenfunctions?

Defintions Let $(X, \mathcal X, \mu, T)$ be a measure preserving system. Let $U_T:L^2_\mu\to L^2_\mu$ be the associated Koopman operator. We will write $\mathcal X_0$ to denote the $\sigma$-algebra ...
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73 views

If $(X\circ\tau^n)_{n\in\mathbb N}$ is $\operatorname P$-independent, then $\operatorname P$ is $\tau$-ergodic

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau$ be a measurable map on $(\Omega,\mathcal A,\operatorname P)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $X:\Omega\...
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1answer
41 views

Show ergodicity of transformation of ergodic map

Let $(\Omega,\mathcal{A},\mathbb{P})$ and $(\Omega',\mathcal{A}',\lambda)$ be two measure spaces. If we have an ergodic map $f\colon \Omega \rightarrow\Omega$ and a bijection $h\colon \Omega'\...
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63 views

Lyapunov exponents: Why do we know that the changes happen at an exponential rate

Let $E$ be a $\mathbb R$-Banach space, $\Omega\subseteq E$ be open, $f:\Omega\to\Omega$ be continuously Fréchet differentiable, $x_0\in\Omega$ and $\varepsilon>0$ with $B_\varepsilon(x_0)\subseteq\...
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121 views

Why is this $L^1$-sequence relatively weakly sequentially compact?

Let $(E,\mathcal E,m)$ be a probability space, $\theta$ be a measurable map on $(E,\mathcal E)$ with $m\circ\theta^{-1}=m$, $s_n$ be a real-valued nonpositive integrable random variable on $(E,\...
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50 views

Why does this sequence of random variables almost surely converge?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $\tau$ be a measurable map on $(\Omega,\mathcal A,\operatorname P)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$. $(Y_n)...
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1answer
49 views

Which probability inequality is applied in this proof?

I'm trying to understand which probability inequality is used at the end of the following proof of Kingman's theorem in Revuz' Markov Chains book. He's considering a probability space with probability ...
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1answer
27 views

How is ergodic measure defined?

I have a clear definition for an ergodic endomorphism, but I can't seem to find a clear definition for an ergodic measure. For example, the Wolfram definition (https://mathworld.wolfram.com/...
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71 views

Proof of the subadditive ergodic theorem

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $$\mathcal ...
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1answer
52 views

Computation of integral on set determined by orbits

Let $(W, \mathcal{A}, P, T)$ be an ergodic, invertible dynamical system, where $P$ is a dynamical system, and let $n(w) = \inf \{ j \in \mathbb{N} : T^j w \in K \}$, where $K$ is an event with ...
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Trying to understand a proof of the Maximal ergodic theorem

Let $(\Omega,\mathfrak A,P)$ be a probability space, $\Theta:\Omega\to\Omega$ be $(\mathfrak A,\mathfrak A)$-measurable with $P=P\circ\Theta^{-1}$ and $$A_n:=\frac1n\sum_{i=0}^{n-1}F\circ\Theta^i\;\;\;...
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52 views

Finite-to-one measurable factor map preserves entropy?

Let $X$ and $Y$ be compact metric spaces with Borel $\sigma$-algebras $\mathcal{B}$ and $\mathcal{C}$. Let $S: X \to X$ and $T: Y \to Y$ be homeomorphisms. Let $\pi: X \to Y$ be a bounded-to-one ...
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1answer
66 views

If $T$ is measure preserving, then $f\mapsto f\circ T$ is an isometry on $L^\infty$

Let $(E,\mathcal E,\mu)$ be a probability space and $T:E\to E$ be $(\mathcal E,\mathcal E)$-measurable with $$T_\ast\mu=\mu\tag1.$$ How can we show that $$\mathcal L^p(\mu)\ni f\mapsto f\circ T\tag2$$ ...
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1answer
54 views

Proof of source coding theorem using asymptotic equipartition property(AEP)

In this wikipedia article, there is a proof given for one of the directions of the Shannon's source coding theorem using the asymptotic equipartition property (AEP). I am unable to follow the proof. ...
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73 views

What is the measure of the function?

Is the $T$-function $f(x) = (x \text{AND} c) + (x^2 \text{OR} c)$, where $c$ is positive integer ergodic in the space $Z_2$ (p-adic numbers)? What is the measure of this function? I am trying to use ...
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125 views

Is this a property of the sine function?

Because of the equidistribution property of $\{n\mod{2\pi}\}_{n\in\mathbb{Z}}$ in $[0,2\pi]$, my intuition tells me that the following statement about the sine function must be true. We can find an ...
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1answer
87 views

Mean ergodic theorem different assumption

Show that the mean ergodic theorem still holds if we replace the assumption that $T$ is an isometry by the assumption that $T$ is contraction, that is $||T f|| ≤ ||f||$ for all $f ∈ H$ where $H$ is ...
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40 views

Space of ergodic measures is $G_\delta$ in space of invariant measures

I'm reading an old paper by Varadarajan (1963) titled "Groups of automorphisms of Borel spaces" and I'm trying to fulfill the details. There are many questions I thought about for quite a long time ...
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1answer
36 views

Understanding of Random Variables generated by Measure Preserving Transformations

I have difficulties to understand this in ergodic theory (approach form proabrbility theory). Usually, an ergodic process is defined by: $$ X_n(\omega) := X(\phi^n(\omega)), $$ where $\phi$ is a ...
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16 views

Formalizing Siegel lattice integral formula

Siegel's lattice integral formula says that for $f \in L^1 (\mathbb R^n-[{0}])$, $$\int_{\mathbb R^n} f(x) dx = \int_{L_n} f^{*}(\lambda)$$ where on the right we integrate over the covolume lattices ...
3
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1answer
130 views

Birkhoff averages convergence

Let be $(X,\mathcal{A},\mu)$ a probability space, $T:X\to X$ a measurable tranformation preserving $\mu$ and $f: X \rightarrow \mathbb C$ a measurable function. Show that for almost every $x \in X$ ...
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1answer
25 views

Lebesgue-continuous Borel probability measures on $[0, 1)$ ergodic with respect to the doubling map $Tx = 2x \mod 1$

I'm learning about unique ergodicity and how a transformation can have different measures with respect to which it's ergodic. I read that, for example, if $T : [0, 1) \to [0, 1)$ is the doubling map $...
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24 views

Possitive Recurrent States And Aperiodic Properties

Any ideas on how to prove the following statement? Let $P$ be an irreducible transition matrix. $\nu$ an initial distribution and $\displaystyle\pi=\lim_{n\rightarrow\infty} \nu P^n$ and denote $A_n:=...
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2answers
86 views

How to describe the dynamics of this gamble?

Suppose you have a $100 and you are offered a chance to play a game involving a fair coin toss: If you throw heads your wealth increases by 50%. If you throw tails your wealth decreases by 40%. ...
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1answer
42 views

Applications of thermodynamic formalism in dynamical systems theory

I'm a senior undergraduate mathematician and physicist. I'm currently engaging in undergraduate research course, and willing to learn what thermodynamic formalism is and how it is applied to ...
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1answer
33 views

Measure preserving action of topological group $G$ on probability space $X$ gives continuous embedding $G \to L^2(X)$?

On page 13 of Bekka and Mayer "Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces" the authors study measure preserving actions of a locally compact topological group $G$ ...

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