# Tagged Questions

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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### T invariant probability measure on a compact space $X$

Let $T:X\to X$ be a continuous map defined on the compact space $X$. I read that if $X$ is a metric space then the set of T-invariant probability measure is not empty. I want to know if this result ...
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### 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 ...
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### 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, ...
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### 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?
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### 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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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 A_1X_0.... 0answers 69 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 ... 3answers 73 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 ... 0answers 61 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 ... 0answers 24 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" ... 1answer 40 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 ... 1answer 43 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 ... 1answer 118 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 ... 4answers 95 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 ... 1answer 77 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_{[0,... 1answer 73 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 (topological)... 1answer 84 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 ... 1answer 55 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} $$exists and has common value. What is this? (no ... 1answer 58 views ### If x=[a_0,a_1,\dots] show that \mu-almost every x \in (0,1/N] is infinitely recurrent Let G be the Gauss map,$$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 Gauss ... 2answers 43 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 ... 1answer 65 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 ... 1answer 35 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 ... 1answer 97 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 ... 1answer 37 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 ... 1answer 41 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 \... 1answer 112 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 ... 1answer 58 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 \...
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 ...