# 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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### What does it mean for a function to be invertible 'almost everywhere'

It seems to me that the correct definition of a measure-theoretic inverse for a function f is a function g such that $f \circ g$ and $f \circ g$ are the identity almost everywhere. The problem I have ...
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### Extension of erdodic theorem with WLLN

Suppose you have a ergodic (or irreducible) Markov chain $(A_t)_{t\geq0}$ in continuous time. denote by $\pi$ the invariant distribution of $A$. If $f$ is a function of $A_s$ which is integrable w.r.t....
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### When is the weak limit of operators invertible?

Suppose $T_i$ are invertible operators in $L^{2}(X)$ for X a Lebesgue Probability Space. Is the following true? 1. If the $T_i$ converge weakly to $S$, then $S$ is not necessarily invertible. ...
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### Weak convergence of finite measure preserving transformations

I am reading King's paper "The commutant is the weak closure of the powers, for rank-1 transformation" and I am not able to show that: (0.1) "If the $T_i$ are invertible measure preserving ...
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### Showing an ergodic toral automorphism is not measurably isomorphic to an ergodic circle rotation

The question as listed in the title is the question statement, only I do not want to use that one is mixing and the other is not. Is it true that measurably isomorphic spaces are either both mixing ...
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### Measure on torus invariant under multiplication

Let $T: [0,1] \rightarrow [0,1]$ be a multiplication by $\beta >1$ mod $1$. Show that $h(x) d x$ is $T$-invariant where $$h (x) = \sum_{n \geq 0} \beta^{-n} \chi _{[0,T^n (1)]} (x)$$ ($\chi$ is ...
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### Hyperbolicity without ergodicity?

I have a question concerning the ergodic properties of hyperbolic Hamiltonian flows. Let $\Phi_{H}^{t}$ be a Hamiltonian flow on a symplectic manifold $\mathcal{M}$. If $\Phi_{H}^{t}$ is Anosov on a ...
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### Hausdorff dimension via ergodic theory

This is definitely a soft question, but it was recently mentioned to me that one can study the dimension of fractals via ergodic methods. I'm familiar with ergodic theory on about the level of ...
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### Exercise 2.1.1 from Einsiedler and Ward

I am studying Ergodic Theory for the first time, and am using the book "Ergodic Theory with a view towards Number Theory" by Einsiedler and Ward. I got stuck at the very first exercise problem, ...
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### Conditioning on invariant sigma algebra with respect to ergodic measure

So this question arose to me while applying the Ergodic theorem. If $X$ is a finite state (in $\{1,\dots,d\}$) continuous-time Markov chain, which is ergodic, then $X$ has a unique invariant ...
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### Converges of measures.

Good afternoon, we have the following: Let $(Y,d)$ is a general metric space, $\mathcal{M}(Y)$ is the set of finite Borel measures on $Y$ and $C_B(Y)$ denotes the Banach space of bounded continuous ...
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### A Question Regarding Markov Chains and Ergodicity

Suppose the Markov chain with Probability Transition Matrix, $P$ = ($p{_x}{_y}$) is ergodic and $p{_m}(x, y) > 0$ for all states $x$ and $y$. If $n ≥ m$, show that $p_n(x, y) > 0$ for all ...
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### Ergodic actions of orthogonal group $O(2)$

I am looking for explicit ergodic action of $O(2)$ on a von Neumann algebra $M$. ($O(2)$=orthogonal group of $2\times 2$ matrix)
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### Approximate eigenvalues of an ergodic invertible transformation

Consider a non-atomic probability space $(X,\mathcal{B}, m)$. Let $T: X \to X$ be an ergodic invertible measure preserving transformation.Let $U_T$ be the Koopman operator associated with $T$. Show ...
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### application of Birkhoff Ergodic Theorem

Let $(\Omega, \mathcal{F} , P)$ be a probability space and let $T$ and $S$ be ergodic, measure-preserving transformations of $(\Omega, \mathcal{F} , P)$. Let $X : \Omega → \mathbb{R}$ be a bounded ...
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### Non-atomic, ergodic measure which is left and right shift invariant.

Given a one-sided shift space, say $X = \prod\limits_{n=1}^\infty \mathbb Z_2$. Denote the left shift by $T$: $T(x_1 x_2 x_3\cdots) = x_2x_3 \cdots$. There are lots of examples of $T$-invariant ...
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### Are $L_\infty$ functions measurable/integrable?

Lemma 2.6 of "Ergodic Theory with a view towards Number Theory" (Einsiedler-Ward) involves: $$\int f d\mu$$ where $f \in L^{\infty}$. Actually it is a calligraphic $L$ and I'd love if you would ...
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### What is known about the space of measure-preserving transformations?

I started reading about measure-preserving transformations, the ergodic theorems and mixing, but I was also wondering what is known about the space of measure-preserving transformations. The books ...
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### If $1$ is a simple eigenvalue of $T$, then $T$ is an ergodic measure-preserving transformation

Let $T : X \to X$ be a measure-preserving transformaton and $U_T : L^2 (X, \mu) \to (X, \mu)$, $$(U_T f) (x) = f(Tx).$$ I have to show that if $1$ is a simple eigenvalue of $U_T$, then $T$ is ergodic....
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### Different ergodic probability measures are mutually singular

Can someone, please, give me a hint on how to demonstrate that different ergodic probability measures are mutually singular? Thank you!
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### Ergodicity of stochastic process

If one can show that the process converges to a stationary process in probability, does it mean that the process is ergodic?
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### Example for a non-ergodic stationary process

Let $(X_n)_{n \in \mathbb{N}}$ be a (strictly) stationary process and let $T$ denote the left-shift on $\mathbb{R}^\mathbb{N}$, i.e. $T((x_n)_{n \in \mathbb{N}}) = (x_{n + 1})_{n \in \mathbb{N}}$. ...
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### Theorem of Daniell-Stone (uniqueness when assuming compactness)

Theorem of Daniell-Stone. Let $L$ be a $\sigma$-continuous abstract integral on a Stone lattice V of real-valued functions on $\Omega$ and let $\mathcal{A}(V)$ denote the set of all $V$-open sets. ...
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### Strengthening Poincaré Recurrence

Let $(X, B, \mu, T)$ be a measure preserving system. For any set $B$ of positive measure, $E = (n \in \Bbb N |\; \mu(B \; \cap \;T^{-n}B) > 0)$ is syndetic. This exercise comes from Einseidler ...
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### Ergodicity of stochastic recursive process

Does anyone know how one can show ergodicity for a recursive stochastic process determined by the following equation: $$X_n = f(\varepsilon_{n-1},Y_{n-1})X_{n-1},$$...
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### Criterium in ergodic theory.

Given a topological space $X$ with a probability measure $\mu$ and a continuous transformation $T:X \rightarrow X$ which preserve measure. If a set $A$ with $1>\mu(A)>0$ is such that the ...
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### When does a flow inherit ergodicity from a Poincare section?

Assume we have a smooth compact manifold $M$ with boundary and a smooth complete vector field $X$ on $M$. Let $\phi^{t}$ be the resulting flow and let $\mu$ be a probability measure on $M$. Define the ...