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
12 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} ...
5
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1answer
47 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 ...
2
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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 ...
2
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0answers
49 views
+50

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 ...
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0answers
10 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
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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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25 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 ...
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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
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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
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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 ...
2
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0answers
46 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
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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
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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 ...
5
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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 ...
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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 ...
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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 ...
2
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1answer
40 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: ...
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.
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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?
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0answers
24 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 ...
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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: ...
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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 ...
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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 ...
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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?
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2answers
43 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 ...
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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
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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
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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?
2
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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. ...
2
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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 $\mathcal{X}$ ...
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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 ...
6
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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 ...
0
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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
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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 ...
1
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1answer
38 views

Question About Definition of Lyapunov Exponents

I just have a quick question about the definition of Lyapunov exponents. My textbook defines them for a smooth map $f:M\to M$, where $M$ is a smooth manifold. For $x\in M$ and $v\in T_xM$: ...
3
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0answers
41 views

Understanding Proof of Poincare Recurrence Theorem

I'm trying to follow a proof in my book of the Poincare Recurrence Theorem, but I have three questions about this proof: Theorem Let $(X,\Sigma,\mu$ be a finite measure space, $f:X\to X$ be a ...
0
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1answer
21 views

Point with a dense trajectory

Let's consider a map $\varphi: [0, 1] \rightarrow [0, 1]$ so that $x \mapsto \{2x \}$. I would like to find a point $x$ so that its trajectory is everywhere dence in $[0,1]$. Firstly, the basic idea ...
2
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1answer
38 views

Is there is a map from the 3-dimensional ball to itself that does not admit an invariant measure?

Krylov-Bogolybov theorem states that if $X$ is metrizable compact space and $f: X \rightarrow X$ is continous then it admits an invariant Borel probability measure. I would like to build a map $F$ ...
0
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0answers
12 views

Can you think of interesting examples of equidistribtuion of a surface measure supported on a curve?

I am trying to understand equidistribution, and am trying to think of two dimensional examples where a measure is supported on a one dimensional subspace (line/curve/...), but in the limit under the ...
1
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1answer
71 views

What probability topics can be read without Measure Theory

I know Intermediate Probability Theory and Statistics (distributions, convergence concepts, characteristic functions, etc.) and am pretty good at these. I would love to know more about Probability ...
3
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1answer
66 views

Orbit of transformation on point in measure space returns to subset

Let $X$ be a measure space and $T:X\to X$ a measure preserving transformation. The Poincare recurrence theorem states that for any $T$ and any $A\subset X$ with $\mu(A) > 0$ (we take $\mu(X)=1$) ...
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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{ ...
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0answers
47 views

Bakers map has dense orbit.

Let $X=[0,1]^2$, $L$-the Lebesgue measure on the $\sigma$-field of Borel sets. Define the map $T:X \to X$: $$T(x,y)= \begin{cases} (2x,y) \textrm{ for } x \in [0,\frac{1}{2}), y \in [0,1] \\ (2x-1, ...
3
votes
1answer
84 views

Definition of topological entropy

What the meaning of the limit that appears in the definition of topological entropy? Let $X$ a compact metric space and $f\colon X\to X$ a continuous function and a subset $K \subset X$. The ...
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1answer
48 views

What is measure theoretical entropy in multidimensional symbolic dynamical systems?

Can any one describe the term entropy used in dynamical systems, and what is roll of entropy in symbolic dynamical systems and please give the brief introduction on measure theoretical entropy?
4
votes
2answers
73 views

Ergodic components of Markov chain by transition matrix

I would like to find an algorithm for obtaining all ergodic components of a finite Markov chain with discrete time defined by its transition matrix (i.e. ergodic subchains into which the given chain ...
2
votes
1answer
36 views

Measure preserving transformation $T([a,b])\subset P$ if $\lambda(P)=\lambda([a,b])$

"Suppose that a measurable subset $P \subset [0,1]$ and the interval $I = [a,b] \subset [0,1]$ are such that $\lambda(P) = \lambda(I)$, where $\lambda$ is the Lebesgue measure on $[0,1]$. Show that ...