# 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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### 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 ?
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### 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 ...
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### 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 $f_{t+1} = F(f_t)$. ...
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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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?
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### $T(x) = \{\frac{1}{x} \}$ 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. ...
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### 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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### 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 ...
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### 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 ...
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### 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 $I(x)=I(f(x))$. ...
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### 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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### 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 ...
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### 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 ...
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### 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 $v\in T_xM$: ...
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### 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 ...
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### 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 ...
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### 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 $F$ ...
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### 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 ...
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### 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 ...
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 $\mu(X)=1$) ...