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.

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 ...
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 ?
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 ...
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 $f_{t+1} = F(f_t)$. ...
0answers
18 views

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 ...
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?
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 ...
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 ...
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 ...
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?
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 ...
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 ...
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. ...
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 ...
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 $\mathcal{X}$ ...
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 ...
1answer
14 views

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 $I(x)=I(f(x))$. ...
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 ...
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 ...
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 ...