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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5
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2answers
57 views

An intuitive explanation of how the mathematical definition of ergodicity implies the layman's interpretation 'all microstates are equally likely'.

I'm self-studying Statistical Mechanics; in it I got Fundamental Postulate of Statistical Mechanics and that took me to ergodic hypothesis. In the most layman's language, it says: In an isolated ...
1
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0answers
19 views

Spectrum of $T^2$

Is it possible that $T^2$ has a discrete spectrum when $T$ is an invertible measure-preserving transformation whose spectrum is continuous or mixed ?
1
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1answer
17 views

can two Markov kernels be close in total variation and differ in their ergodicity properties?

This is a question inspired by a recent MCMC paper and linked to an earlier question of Sam Livingstone that did not get any answer. Given two Markov kernels $\mathfrak{K}$ and $\mathfrak{H}$ such ...
2
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0answers
18 views

Ergodicity of the natural measure implies uniqueness of the invariant density?

Consider a dynamical system $x_{t+1} = F(x_t)$ defined in $\Omega$ and its natural measure to be $\mu$. The Perron-Frobenius operator $F$ maps the density $f$ in time according to $f_{t+1} = F(f_t)$. ...
1
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0answers
18 views

Why does ergodicity not neccesarily imply ergodic for the mean?

I'm trying to answer a question where I have an ergodic and covariance stationary process $\{x_t\}$, and without imposing further moment conditions need to prove $\frac{1}{n} \sum\limits_{t=1}^n x_t^2 ...
0
votes
1answer
33 views

Ultrafilter and upper natural densities

It is straightforward to show that there is an ultrafilter $\mathcal{U}_0$ on the positive integers such that every element $A\in \mathcal{U}_0$ satisfies $$d^\ast(A):=\limsup_{n\to +\infty} ...
3
votes
0answers
83 views

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers ...
2
votes
0answers
23 views

Moving Average of an Ergodic Markov processes.

Let $ \{X(t); t\geq 0\} $ be an ergodic Markov process, and let G be a positive integrable function with $\int_0^\infty G(x)dx=1$. Does $$F(t)\doteq G*X(t) = \int_0^t G(t-s)X(s)ds $$ converge in ...
1
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0answers
11 views

Question about a type of continuous state Markov-process.

I was wondering if anyone knows whether a particular result has been proven or is indeed true. My problem is as follows. Suppose I have a stationary, ergodic Markov chain that follows a process of ...
0
votes
1answer
29 views

multiples of subset of $\mathbb{N}$

Suppose that $ A_1\cup \dotsm A_n=\mathbb{N}$ is a partition of $\mathbb{N}$ into disjoint subsets. Is it true that there is an integer $1 \leq k \leq n$ such that the set $A_k\cap 2A_k$ is infinite?
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0answers
26 views

Bernoulli-Shift is a stationary process

Let be $(\Omega, \mathcal{F}, \mathbb{P})=([0,1), \mathcal{B}([0,1)), \lambda)$. We define $Y_n : \Omega \rightarrow \Omega$ by $Y_n := 2Y_{n-1} \mod 1$. Many sources claim that this is a stationary ...
1
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0answers
17 views

Invariant $\sigma-$ field of double infinite stationary process

We know any stationary process ${(X_n)}_{n \in \mathbb{Z}}$ can be represented as $X_n(\omega)=X(\phi^n\omega)$. where we take the shift $\phi$ on the canonical space of the process and $X$ maps a ...
0
votes
0answers
22 views

Non-integrability and splitting of separatrices

It is well-known that the (first-order) Melnikov method is the standard technique to detect non-integrability of a perturbed system of ordinary differential equations or maps. Namely, the unperturbed ...
2
votes
0answers
47 views

Convergence of empirical average of Markov chain from transient class

I am trying to get an intuition of how to understand the limit of the empirical average $$\frac1n\sum_{i=1}^nX_i\tag{$\ast$}$$ of some Markov chain $(X_n)_n$ with transition matrix $P$ (let's assume ...
2
votes
0answers
16 views

Deterministic time changed ergodic process

This is more of a "ask-for-idea" than "ask-for-answer" question: Suppose $\{X_t\}$ is an ergodic process with a known stationary\limiting distribution $\pi$. Let $f(t)$ be a deterministic and ...
5
votes
1answer
57 views

Sawyer's proof that a certain lim sup of sets has full measure

Given $(X, \mathcal{A}, \mu)$ a probability space, let $\mathcal{F}$ be a family of $\mu$-invariant measurable functions, closed under composition, with the following property: If $A$ is a measurable ...
3
votes
2answers
539 views

The fractional parts of the powers of the golden ratio are not equidistributed in [0,1]

Let $$a_n=\left(\frac{1+\sqrt{5}}{2}\right)^n.$$ For a real number $r$, denote by $\langle r\rangle$ the fractional part of $r$. Why is the sequence $$\langle a_n\rangle$$ not equidistributed in ...
2
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0answers
16 views

Is there a mixing condition to get the decay property I want?

Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation. Consider a measurable set $A\subset X$ with $0<\mu(A)<1$ For each $N$ define ...
4
votes
2answers
42 views

Minimal distance for Irrational Rotations on the circle

It is a well known fact that for $\alpha$ irrational that $\langle n\alpha\rangle$ is dense on the unit circle. I want to know what the result is for computing $a_n=\min_{1\le N \le n} |\langle N ...
1
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0answers
85 views

Testing whether a particular set of measures borelianas is a set of Baire

Let $X$ compact metric space and $F:X\times \mathbb{R}\rightarrow X$ flow continuous ($F(x,t)=F_t(x)$). If $\delta>0$ we define $$\Lambda(x,\delta)=\bigcup_{h\in ...
12
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10answers
2k views

Reference for Ergodic Theory

I am looking for a good introductory book on ergodic theory. Can someone recommend some introductory texts on that?
3
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2answers
113 views

Reference request : Ergodic theory and Number theory

I would like to work on relation between Ergodic theory (or Dynamical systems) and Number theory but I am looking for a good reference book, Lecture note and also I like to get familiar with Articles, ...
0
votes
0answers
26 views

expected value of a doubly stochastic matrix with i.i.d entries

I am now thinking a problem: what is the expected value of doubly stochastic matrix with i.i.d entries Each entries is i.i.d in $[0,1]$. Will the answer be a matrix with all entries ...
4
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0answers
50 views

Equidistribution theorem of Weyl

Have you examples of applications of Equidistribution theorem of Weyl in proofs of irrationality of numbers? I don't know if "if and only if" is true for this theorem.
1
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2answers
45 views

Can a mixing process be non-stationary?

I was always under the impression that a mixing process is ergodic and an ergodic process is necessarily stationary, so that a mixing process is stationary. I have come across a paper discussing ...
2
votes
1answer
41 views

Cesaro summation

Let's consider $\{a_{n} \} -$ a bounded sequence of real numbers. Is it true that $\frac{1}{n} \sum_{k=0}^{n-1}{|a_{k}|^{p}}$ (Cesaro sums) converges or diverges for all $p \geq 1$? (more presicely: ...
0
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0answers
7 views

Varying measures and invariant measures

Are there any results related to varying measures? Why are invariant measures so useful/important?
0
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0answers
25 views

Ergodic means and Birkhoff theorem

Let's consider the following map $$F(x, y) = \lim_{n \to \infty}{\frac{1}{n} \sum_{k=0}^{n-1}{f(\{x + ky \})}}$$ and $f(x) = x(1-x)$. I would like to evaluate the value of $F(x, y)$ for arbitrary ...
1
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0answers
15 views

Confusion about the concept of ergodicity in random process

This is a basic question I came across when I started learning random processes. Suppose I have a random variable $f(t) = G(t) + n(t)$ where $G(t) = A\exp(-t^2/T^2)$ (i.e. a deterministic function ...
3
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0answers
22 views

Reference on the history of ergodic theory

I'm looking for some good books on the history of ergodic theory. I'm a Ph.D student in the field, and I am taking Steven Weinberg's advice to learn about the history of my field: ...
1
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0answers
19 views

Strong mixing property for endomorphisms of a finite set

Let's consider $X = \{1, 2, \ldots, n \}$. I would like to establish, how many of the maps $f: X \to X$ have the following strong mixing property: For a given triple $(X, \mathcal{B}, \mu)$, $T: X ...
11
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0answers
769 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 ...
0
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1answer
73 views

Eventually periodic continued fraction implies root of polynomial of degree 2

How to prove that every irrational number with eventually periodic continued fraction expansion is a root of a polynomial of degree 2?
0
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1answer
23 views

Why is the set $E=\{x\in X \:| \exists N \in \mathbb N \forall n>N d(T^n x,x) \geq \epsilon\} $ measurable?

I'm trying to prove a theorem in Ergodic Theory, in which I want to be able to use a set being measureable, but I don't find it too easy to understand why it is. Wounder if you could help. Let ...
0
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0answers
37 views

An aquidistributed sequence

Prove that $\{an^\sigma\}$ is equidistributed in $ [0,1) $,if $\sigma>0$ is noninteger and $a\neq 0$. I know how to solve this problem if $\sigma <1$ , so it is not a duplicate of ...
2
votes
0answers
23 views

Ergodicity under measure-theoretic isomorphism

Suppose we have two measurable dynamical systems $(X_1,\mathcal{B}_1,\mu_1,T_1)$ and $(X_2,\mathcal{B}_1,\mu_2,T_2)$, with $\mu_i(X_i)=1,\ i=1,2$. Suppose they are measure-theoretically isomorphic ...
2
votes
2answers
33 views

why topological conjugacy does not preserve ergodicity?

I want to know why topological conjugacy does not transfer ergodicity from one dynamical system to another?and if we change the maps from continous to C1-differentiable will the problem solve or not?
3
votes
1answer
243 views

Proving Borel Cantelli Lemma using Martingales

I need a hint for exercise 5.2.1 in the book: Ergodic Theory: with a view towards Number Theory By Manfred Leopold Einsiedler, Thomas Ward. In the chapter 5 the authors gives the martingale ...
6
votes
2answers
66 views

A discontinuous almost everywhere map does not admit an invariant measure

Let's consider a map $T: X \rightarrow X$ so that it's discontinuous almost everywhere (in particular, let $X = \mathbb{R}$, and $T = 1_{\mathbb{Q}}$ -- Dirichlet function). Is it true that $T$ does ...
2
votes
1answer
35 views

$T(x) = \{\frac{1}{x} \}$ invariant measure

Let's consider $T: (0,\hspace{-0.05 in}1] \rightarrow [0,\hspace{-0.05 in}1)$ which is defined as $Tx = \{ \frac{1}{x} \}$. It can be shown that Lebesgue measure is not invariant under this map. ...
3
votes
2answers
140 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 ...
2
votes
0answers
25 views

Ergodic theorem in two variables?

I just started learning ergodic theorem due to the need in a research project. I am aware of the following form of ergodic theorem: If $\{X_n\}$ is an ergodic process with state space $\mathcal{X}$ ...
1
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0answers
35 views

Uniform integrability of functional of an ergodic Markov Chain?

Suppose $\{X_n\}$ is an ergodic Markov Chain with general state space $\mathcal{X}$ and stationary distribution $\pi$. Suppose $\forall x\in\mathcal{X}, f(x)\geq 0$ and ...
1
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1answer
14 views

Is there a way to find $\limsup$ and $\liminf$ for ergodic processes almost surely?

Let $\{X_k\}$ be an ergodic process. I know that if $f$ is a smooth real valued function then by Birkoff's ergodic theorem, $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n f(X_k)=\mathbb{E}(f(X_1))\mbox{ ...
5
votes
1answer
111 views

Show that Brownian motion on the unit circle is exponentially ergodic and has the uniform measure as its invariant distribution.

My search results keep bring up planar Brownian motion on the unit disk. However, I am specifically referring to $e^{jW_{t}} = [\cos(W_t),\sin(W_t)]^{T}$ where $W_t$ is Brownian motion. I am at a ...
5
votes
3answers
399 views

The prime numbers do not satisfies Benford's law

A set of numbers is said to satisfy Benford's law if the leading digit d (d ∈ {1, ..., 9}) occurs with probability, $$ P(d)=\log_{10}(d+1)-\log_{10}d,$$ how do you prove that the prime numbers do ...
0
votes
1answer
20 views

Integrability and area-preservation property of maps

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. ...
0
votes
1answer
40 views

Rotation map on $S^1$ preserves measure

I'm having a little trouble understanding following the example in my book as to why the rotation map $R_{\alpha}$ preserves Lebesgue measure. We have $R_{\alpha}([x])=[x+\alpha]$ and ...
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0answers
30 views

Question about dense trajectory on $k$-dimensional torus under rotation map

Today when I was doing ergodic theory problems I faced with following problem: Assume rotation map on $k$-dimensional torus under $\alpha=(\alpha_1,...,\alpha_n)$ then orbit of all $x$ in ...
1
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1answer
29 views

Computing Lyapunov Exponents

Consider $\mathcal{T}=S^{1}\times D^{2}$, where $S^{1}=[0,1]\mod 1$ and $D^{2}=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}\le 1\}$. Fix $\lambda\in(0,\frac{1}{2})$ and define the map ...