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
9 views

If a Borel $\sigma$-algebra is genertated by a collection, then the collection is closed under intersections?

The question is: If $\sigma\left(\tau\right)$ is a Borel $\sigma$-algebra such that is genertated by a collection $\mathcal{B}\subseteq\sigma\left(\tau\right)$, then $\mathcal{B}$ is closed under ...
1
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1answer
21 views

¿If a Borel $\sigma$-algebra is generated by a collection of subsets of algebra, then the Borel $\sigma$-algebra is generated by the algebra?

Let $\left(M,\sigma\left(\tau\right)\right)$ a measure space with $\sigma\left(\tau\right)$ a Borel $\sigma$-algebra where $\tau$ is a topology in $M$. Suppose there is a algebra $\Gamma$ in $M$ ...
2
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1answer
27 views

Relation between the density function (measure theory) and density (physics)

I was reading some notes on Ergodic Theory and there is this sentence: Suppose we distribute mass on $X$ according to the mass density $fd\mu$, $f \in L^1(\mu)$,$ f \geq 0$, and then apply $T$ ...
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0answers
38 views

How can we construct Markov partitions for Smale Horseshoe and Solenoid?

I was reading the book Ergodic Theory, Hyperbolic Dynamics and Dimension Theory by Luis Barreira. In Chapter 7, it is asked to construct Markov partitions for the Smale Horseshoe and the solenoid. In ...
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0answers
31 views

Topologizing ergodic system so that certain function becomes continuous

Let $X$ be a compact metric space and $\mathcal{B}$ its Borel $\sigma$-algebra. Suppose that $(X,\mathcal{B},\mu,T)$ is an invertible ergodic system ($T$ is only a measurable isomorphism, not ...
2
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1answer
58 views

$\mu$ is a $f$-invariant measure

Let $\left(M,\sigma\left(\tau\right),\mu\right)$ a measure space where $\mu$ is a measure finite, $\tau$ is a topology in $M$, i, e, $\sigma\left(\tau\right)$ is a Borel $\sigma-$algebra. Let ...
2
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2answers
31 views

Tent map invariant density

Is there a formula for the invariant density for the tent map $f_t$ (for $\sqrt{2}\leq t\leq 2$)? $$ f_t = \begin{cases} tx, & 0\leq x<1/2 \\ t-tx, & 1/2 \leq x\leq 1. \end{cases} $$ By ...
2
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1answer
34 views

Reference request : Ergodic theory and Number theory

I would like to work on relation between Ergodic theory(Or Dynamical system ) and Number theory but I am looking for a good reference book, Lecture note and Also I like to get familiar with Articles, ...
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0answers
25 views

Notation $\mathrm{mod} $ in ergodic theory

Does someone know, what exactly is meant by the following: $$T^{-1}A=A \mod \mu$$ where $\mu$ is a $T$-invariant measure?
3
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1answer
28 views

Krylov-Bogoliubov Theorem

I've just started to learn for my ergodic theory exam and have the following question, because I can't find anything in my notes: What happens, if the space X is not compact or more specifically what ...
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0answers
24 views

Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have $$X_n = A_n \cdots ...
2
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0answers
29 views

Over a compact space, the set of continuous functions are everywhere dense in the set of all measurable functions

WARNING: The following post certainly contains too much information. I've included the complete context of my question so that it is clear what I'm talking about and where my question comes from. Feel ...
4
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3answers
63 views

Topologizing Borel space so that certain functions become continuous

Let $X$ and $Y$ be compact metric spaces. Let $f:X \to Y$ be a Borel measurable map and suppose that $T:X \to X$ is a homeomorphism. Can one change the topology on $X$ such that $X$ is still a ...
2
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0answers
39 views

Unique Ergodicity

Show that unique ergodicity is a topological invariant. Is arguing as follows an overkill (hopefully if the logic is correct --- I have a feeling that there has to be a way a T-invariant measure has ...
1
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0answers
16 views

nice stationary process with discrete spectrum

I'd like to have an example of a stationary process $(X_n)_{n \in \mathbb{Z}}$ on a finite alphabet for which the shift $T$ is ergodic and has discrete spectrum, and for which there is a "nice" ...
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0answers
24 views

Positive upper density implies positive upper density of intersection with shift.

The following problem appears in [1]: 2.2.2(a) Use Exercise 2.2.1 to show the following. If $A \subseteq \mathbb{N}$ has positive density, meaning that $\mathbf{d}(A) = \displaystyle \lim_{k ...
2
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1answer
31 views

Almost every $x$ is a cluster point of its own trajectory.

The following problem appears in [1]: 2.3.2(a) Let $(X, d)$ be a compact metric space and let $T:X \rightarrow X$ be a continuous map. Suppose that $\mu$ is a $T$-invariant probability measure ...
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1answer
31 views

How long does it take two identical hidden Markov models run on same observations to forget their initial distributions (if ever)?

Let $H_1$ and $H_2$ be two instances of a finite Hidden Markov Model (HMM) $H$. That is, $H_1$ and $H_2$ have identical state spaces $Q$ as well as identical transition $A$ and emission probabilities ...
4
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1answer
52 views

Strongly mixing uniquely ergodic dynamical system

I'm looking for an example of a dynamical system which is both (measure-theoretically) strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are ...
2
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4answers
79 views

Can one generate a sequence of natural numbers whose density has a given distribution?

Suppose $\{ p_{k} \}$ is a collection of real numbers with the following properties: 1) $p_k \in (0,1)$ $~~~~$(i.e. $0$ and $1$ are not allowed values) 2) $\sum_{k=1}^{\infty} p_k =1$ An ...
3
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1answer
64 views

Prove existence of Borel set related to the function $f(x)=2x \mod 1$

Let $I=[0,1)$ and $f(x)=2x \mod 1$. Prove that for every $\epsilon>0$ there is $E\subset I$ Borel set s.a $m(I/E)<\epsilon$ and $\lim_{N\to\infty} \sup \left\{|\frac{1}{N}\sum_{j=0}^{N-1} ...
1
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1answer
40 views

why a minimal dynamical system is a ergodic measure-preserving system?

A dynamical system(DS) is a map $(X,T)$ where $X$ is a compact metric space and $T:X-->X$ is a continuous transformation. A minimal DS means for any point $x$ belongs to $X$, $x$ is a ...
2
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1answer
53 views

if $f$ is weakly mixing then $f^n$ is ergodic?

if $f$ is weakly mixing then $f^n$ is ergodic?I think this is false but I cant find a counter example because I dont know transformations weakly mixing but not mixing.can you prove or give a ...
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1answer
46 views

Show that $\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $ using Birkhoff Ergodic Theorem

Show that for Lebesgue-almost every $x \in [0,1)$, the geometric mean $$\lim_{n \rightarrow \infty} \left(\prod_{i=1}^{n} (a_i+1) \right)^{1/n} $$ exists and has common value. What is this? (no ...
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1answer
39 views

If $x=[a_0,a_1,\dots]$ show that $\mu$-almost every $x \in (0,1/N]$ is infinitely recurrent

Let $G$ be the Gauss map, $$G(x)= \begin{cases} 0 & \text{if} \ x=0 \\ \{\frac{1}{x} \}=\frac{1}{x} \ \mathrm{mod} \ 1 & \text{if $0<x\leq 1$}\end{cases}$$ and $\mu$ be the ...
1
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2answers
35 views

Show that $ \lim_{n \rightarrow \infty} \frac{|f(T^n(x))|}{n}=0 $

Let $T: (X, \mathcal{A},\mu) \rightarrow (X, \mathcal{A},\mu)$ be ergodic wrt a measure $\mu$ on $(X,\mathcal{A})$. Show that for any $f \in L^1(X,\mathcal{A})$ and $\mu$-almost every $x \in X$ we ...
0
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1answer
55 views

Show that $Per_n(f)$ of periodic points of period $n$ is finite

Prove that if $f: X \rightarrow X$ is an expansive topological dynamical system of a compact dynamical system $X$, then the set $Per_n(f)$ of periodic points of period $n$ is finite. Any ideas of how ...
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1answer
34 views

A polynomial equality for the square of a self-adjoint positive contraction in $L^2$ — from Krengel's book Ergodic theorems

Another mystery from Ulrich Krengel's textbook - Ergodic Theorems (first mystery). This time it's from page 190, in the proof of theorem 2.7. He takes $P=T^2$, where $T$ is a self-adjoint positive ...
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1answer
38 views

Ergodicity of irrational rotation

It is a well-known fact that the irrational rotation on $S^1$ is ergodic with respect to Lebesgue measure. But each proof I have seen uses Fourier Analysis. Now, Can someone give a proof without ...
2
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1answer
27 views

Variation on ergodic estimates

Let the sequence of random variables $\{X_{n}\}, n = 1,2, \ldots$ be a Markov chain, which is sufficiently "Ergodic" so that it has stationary distribution $\pi$ and for a function $f$ the sequence of ...
1
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1answer
34 views

The map Ti=i+1 mod N is uniquely ergodic

I have a set $X=\{1,2,...,N\}$ and the map $T:X \to X$: $Ti=i+1 \text{ mod } N$. Now I want to show that $T$ is uniquely ergodic and find the unique measure. I know it holds that $T^Nx=x$ iff ...
2
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1answer
76 views

a question on quasi-invariant measures (with respect to the irrational rotations) on the unit circle

Fix a $\sigma$-finite atom-less measure $\mu$ on the unit circle, which is quasi-invariant and ergodic under the rotation $T$ of the angle $2\pi\theta$, $\theta$ irrational. By a well-known result of ...
1
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1answer
23 views

Understanding the proof of an Ergodic theorem for Markov chains

An ergodic theorem for Markov chains is as follows. If a Markov chain $(X_n)_{n \ge 0}$ is irreducible and has an invariant distribution $\pi$, then $$\frac{1}{n} \sum_{k=0}^{n-1} f(X_k) \to ...
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1answer
44 views

Lemma 1.7 in chapter 5 in Ergodic Theorems by Ulrich Krengel.

The lemma is on page 181. Here's the lemma and its proof: I don't understand how did they get: $$\lim | S^{n_k+1}f(\omega)|\leq \gamma (\alpha - \epsilon)+(1-\gamma)\alpha$$ The term that is ...
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0answers
32 views

Cesaro bounded.

The exercise is from Ulrich Krengel's book, Ergodic Theorems, on pages 173-174. First preliminary notions: a function $h$ with $T^*h=h$ is called harmonic, where $T$ is a contraction in $L_1$. $Y= ...
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0answers
52 views

Entropy of beta-expansion

We have the transformation $T: [0,1) \rightarrow [0,1)$ given by $Tx = \beta x \text{ mod } 1$ with $\beta = \frac{1+ \sqrt{5}}{2}$. Calculate the entropy $h_{\mu}(T)$ of $T$ wrt the invariant ...
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0answers
38 views

Krengel's ergodic theorem

I am not sure if this is the correct forum for my present question. Returning to it after a rather long period and hence, have forgotten the conventions. I am unable to understand the proof of ...
5
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0answers
75 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
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1answer
29 views

Is the average of a dense orbit ergodic for shift function?

Let $\sigma$ be the shift function in the space of two-sided infinite sequences of $\{0,1\}$, $X=\{0,1\}^\mathbb{Z}$ equipped with product topology. We know that there is some point $x\in X$ with ...
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0answers
20 views

Summation methods and entropy

I am aware of the theory of divergent series, but don't know much of it. If you have a text to recommend, I'd be glad to hear it. Suppose I have an infinite-dimensional probability vector $\mathbf{p} ...
4
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1answer
43 views

Do primes modulo k form a normal sequence?

For some $k>2$, form a sequence whose nth term is the nth prime that is not a divisor of $k$ modulo $k$. e.g. for $k=4$ the sequence would be 1,3,1,3,3,1,1,3,3,1,3,1... Is this sequence normal, ...
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0answers
39 views

Why Gaussian process is not Ergodic in general?

Can anyone use a simple way to explain this? I heard this in class but I do not know why. By Wiki: a random process is ergodic if its statistical properties can be deduced from a ...
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1answer
32 views

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic.

Prove that the toral endomorphism $T_A: \mathbb{R}^d / \mathbb{Z}^d \rightarrow \mathbb{R}^d / \mathbb{Z}^d$ is not ergodic. A is an integer matrix such that A has an eigenvalue which is a ...
0
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1answer
39 views

Showing that Lebesgue measure is preserved by translations of the $d$-dimensional torus

Let $\underline{\alpha}=(\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$. Show that the transformation $R_{\underline{\alpha}}=\mathbb{T}^d \rightarrow \mathbb{T}^d$ defined by ...
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0answers
29 views

Transitive Actions, Primitive Actions, and Ergodicity

A group action is transitive iff it has one orbit. Intuitively, this seems to say $G$ shuffles around all the elements of the $G$-set. A group action is primitive iff it has no nontrivial blocks, ...
2
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1answer
96 views

The measure generated by the Cantor staircase and the intersection of the Cantor set with its translate

Suppose that $T$ is the shift $\bmod 1$ of the Cantor set by an irrational number $\alpha\in (0,1)$. Consider the measure $\mu$ on the interval $[0,1]$ generated by the Cantor staircase. I'd like to ...
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1answer
29 views

shift power mod 1 of the cantor set by an irrational number and their intersections

Let $C$ be the Cantor ternary set and consider the shift $T_a$ mod 1 of the interval $[0,1]$ for an irrational number $a\in[0,1]$. I'm wondering whether $T_a^k(C)\cap T_a^l(C)=\emptyset$, $k,l\in ...
0
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1answer
20 views

A sequence of numbers' question. (From Krengel's book on Ergodic Theorems).

On page 136 of Ergodic Theorem's by Ulrich Krengel, in the proof he sets: $c_n= \sup_j (n)^{-1} \sum_{i=0}^{n-1}x_{i+j}$ then he argues that for any $k,m$ positive integers one has: $$c_{km}\leq c_m$$ ...
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1answer
49 views

Normal in base 10

We consider a decimal expansion $x=\sum^{\infty}_{i=1} \frac{d_i}{10^i}$ for $x \in [0,1)$. This expansion is generated by map $Tx=10x ($mod $ 1)$ defined on $([0,1), \cal B,\lambda)$ with $\lambda$ ...
1
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1answer
67 views

Induced sytem ergodic implies normal sytem ergodic

Okay, we consider a measure preserving system $(X, \mathcal F, \mu, T)$ and let $A \in\mathcal F$ be such that $\mu(A) > 0$ and $\mu ( \cup ^{\infty}_{n=1} T^{-n}A) = 1 $. Now I want to show that ...