# Questions tagged [ergodic-theory]

Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.

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### A counterexample for ergodic theorem in $L^{p}$ ($0<p<1$)

This may be a similar question of
A counterexample for the Mean Ergodic Theorem in $L_\infty$
where it considers $L^{\infty}$ space.
My question is: can we claim: $\Omega=[0,1[$, endowed with the ...

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68 views

### Null preserving transformation

Suppose that $(\Omega,\mu)$ is a measure space. Let $\tau:\Omega\to\Omega$ is a measurable map such that $\mu\circ\tau^{-1}<<\mu$. Then $\tau$ s said to be null preserving. I want to prove the ...

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119 views

### von Neumann ergodic theorem for $L_p$

Let $\tau:\Omega\to \Omega$ be a measure-preserving transformation with $\mu(\Omega)<\infty$. Define $T:L_p(\Omega)\to L_p(\Omega)$ as $Tf:=f\circ \tau$. I want to prove that for all $1\leq p<\...

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106 views

### Ergodic action on unitary group

Let $G$ be a locally compact Hausdorff group. Assume that $\theta:G\to U_d$ is a group homomorphism where $U_d$ is a finite dimensional unitary group. Consider a action of $G$ on $U_d$ by $g.u:=\theta(...

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### Definition of generic point

I am trying to read a paper named D.S. Ornstein, B. Weiss, Subsequence ergodic theorems for amenable groups, Israel J. Math. 79 (1) (1992) 113–127, doi:10.1007/BF02764805. In this paper the authors ...

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111 views

### G-abelian systems

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state.
Consider the covariant GNS ...

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### Counting simple closed curves

I'm currently trying to understand how to count simple closed curves. I've been reading Alex Wright's survey (https://arxiv.org/pdf/1905.01753.pdf). However, I don't feel like I'm getting the big ...

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302 views

### Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces

Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...

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127 views

### Properties of the spectrum of the Koopman representation

Let $G$ be a discrete countable infinite group acting on a compact metric space $X$ via homeomorphisms preserving a probability measure $\mu$.
A function $\lambda\colon G\to \mathbb C$ is an ...

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168 views

### A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$

This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there or here.
$G$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie ...

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184 views

### Analog of the Birkhoff's ergodic theorem for the sequence of squares

Consider a dynamical system $(X, \mathcal{B}(X), \mu, T)$ where $(X, \mathcal{B}(X), \mu)$ is a measure space and $T$ is a measure-preserving, invertible transformation.
Then by the classical ...

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### Locally compact Polish groups acting on standard Lebesgue spaces

If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, ...

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53 views

### Reduce ergodicity to the ergodicity of the coordinate process

Let $(E,\mathcal E,\lambda)$ be a probability space and $\lambda$ be a measurable map on $(E,\mathcal E)$ with $\lambda\circ\tau^{-1}=\lambda$.
I would like to show that $\tau$ is $\lambda$-ergodic ...

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118 views

### Central limit theorem versus entropy in dynamical systems context

A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...

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### Lyapunov indices of a product of operators

The deterministic part of the proof of the multiplicative ergodic theorem can be proven using Proposition 1.3 in the paper Lyapunov indices of a product of random matrices.$^1$
They consider a ...

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92 views

### Physical measures that are not SRB

It is quite easy to construct a dynamical system which has a physical measure with a positive Lyapunov exponent and zero entropy, just a figure $\infty$ system. By Pesin's entropy formula such a ...

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173 views

### Seeking to understand meaning of “von Neumann spectrum” in a paper of Bader–Furman–Shaker

In attempting to understand the paper "Superrigidity, Weyl groups, and actions on the circle" of Uri Bader, Alex Furman and Ali Shaker (linked at Furman's page)
I find that towards the end of the ...

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139 views

### A subadditive maximal ergodic theorem

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\...

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### For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded?
I feel that it is not easy to treat every irrational $x$.
I have asked in S.E. ...

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269 views

### Why are we interested in operators that share a basis of eigenfunctions?

I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context.
I ...

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74 views

### Factor map between subshifts preserving topological pressure (or measure-theoretic entropy)

Let $G$ be a countable amenable group and let $X,Y$ be subshifts with finite alphabet over $G$. Suppose that $h(X) = h(Y)$ (equal topological entropy). I am interested in continuous factor maps $\pi: ...

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72 views

### Invariant subspaces of Markov operators

I am currently working on some kind of graph theoretic problem and the following question came up:
Suppose you have a Markov operator $T$ on $\ell^\infty$, that is a positive, bounded operator such ...

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81 views

### Existence of large first return times

Let $(X,T,\mu)$ be a measure preserving system, with $\mu$ a probability measure. Let $E \subset X$ of positive measure and $\tau_E$ be the first return time to $E$. Then the Kac Lemma asserts that $$\...

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75 views

### Applying the Abramov-Rokhlin skew product entropy formula to a bounded-to-one factor

Let $(X, \mathcal{B}, \mu, S)$ and $(Y, \mathcal{C}, \nu, T)$ be invertible probability-measure-preserving systems, with a measurable factor map $\pi: X \to Y$, i.e. $\pi \circ S = T \circ \pi$. ...

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49 views

### More formulas for joint entropy and for trace form entropies

Linked to some applications of entropy to combinatorics I'm looking for formulas expressing the joint entropy of two r. v. as a function of the conditional entropy . For example
For BWS extensive ...

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101 views

### Quantum ergodicity of Eisenstein series on arithmetic quotients of hyperbolic space

Let $E(z,1/2+it)$ be the Eisenstein series furnishing the continuous spectrum of the Laplace operator $\Delta$ on $X=PSL_2(\mathbb{Z})\setminus H^2$ and $dV(z)=y^{-2} \,dx \,dy$ be the volume element ...

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79 views

### Gurevich's entropy and topological entropy in a countable Markov shift

Good afternoon, I understand that Gurevich's entropy and topological entropy coincide when the countable Markov shift is topologically mixing (right?)
Does anyone know of an example or a reference ...

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### Possible Birkhoff spectra for irrational rotations

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...

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81 views

### Ergodicity of induced system

Suppose $(X,\mathcal{F},\mu,T)$ is an ergodic measure preserving dynamical system.
Let $Y\subset X$ be such that $\mu(Y)>0$ and suppose there is an integrable function $R:Y\to \mathbb{N}$ such that ...

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170 views

### Fully invariant measures for rational functions

Let $f(z)$ be a rational function of degree $d \geq 2$, with complex coefficients. I am interested in fully invariant measures for the dynamical system $(\mathbb C_\infty,f)$, where $\mathbb C_\infty$ ...

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82 views

### Ergodic automorphism is mixing of all orders

I'm trying to solve an exercise in "Ergodic Theory with a view towards Number Theory".
The exercise is suppose $T$ is an ergodic automorphism on an compact abelian group, show that it is mixing of ...

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### Is a Hilbert space valued processes weak stationary, ergodic if it's nonanticipative w.r.t. a weak stationary, ergodic process?

Let $(\varepsilon_k)_{k\in\mathbb{Z}}$ be a weak stationary, ergodic process with values in a Hilbert space $\mathcal{H}$ and let $(X_k)_{k\in\mathbb{Z}}$ be another process with values in $\mathcal{H}...

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### Smooth dynamics with zero Lyapunov exponents

Apologies if this is a vague question.
It seems that a lot of the literature in smooth dynamics is focused on understanding systems that exhibit hyperbolic/non-uniformly hyperbolic behavior. In other ...

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112 views

### Topological weak mixing vs measure-theoretic weak mixing

Let $X$ be a compact metric space and $T$ a continuous map from $X$ to $X$. The system $(X,T)$ is called topologically weakly mixing if the product system $(X\times X,T\times T)$ is topologically ...

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72 views

### Examples of minimal topological systems which are not intrinsically ergodic

Consider dynamical systems $(X,T)$ where $X$ is a compact metric space, $T:X\rightarrow X$ is continuous, the system is minimal and finally, $0<h_{\rm{top}}(X)<\infty$. I am looking for examples ...

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### Is there a term for a not-necessarily-convex set whose non-extreme points can be expressed as a linear combination of two other points in the set?

This question was asked on Math.SE here, but received no replies after several months. So I have posted it here, though with somewhat revised structuring of the question.
Let $V$ be a real vector ...

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52 views

### Measure invariant under circle maps

Consider continuous bijections (may even assume these are homeomorphisms or diffeomorphisms if it helps) from the circle onto itself given by $x \mapsto x + s_i(x)$ where $i = 1,2$ or $3$. (I'm ...

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79 views

### Entropy spectrum is not concave

Let $T:[0, 1]\rightarrow [0, 1]$ be map such that $T(x)=4x(1-x)$. For any $\alpha \in \mathbb{R}$, we define the level set as follows
$$F(\alpha)=\{x\in [0,1]: \lim_{n\rightarrow \infty}\frac{1}{n}\...

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### Have “measures that are invariant on a non-invariant sub-$\sigma$-algebra” been studied before?

Definition. Given a measurable space $(X,\mathcal{X})$, a measurable map $f \colon X \to X$, and a sub-$\sigma$-algebra $\mathcal{Y}$ of $\mathcal{X}$, I will say that a probability measure $\mu$ on $(...

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317 views

### Help with definition of Liouville measure

$\require{AMScd}$For a Riemannian manifold $M$, I have read authors talking about a 'Liouville measure' on the unit tangent bundle $\operatorname{T}^1(M)$ and then proceed to claim/prove that it is ...

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115 views

### Uniform upper bound on contraction coefficient w.r.t total-variation metric, of a certain set of block-diagonal Markov kernels

Disclaimer. This is related to another question I've asked on the TCS site https://cstheory.stackexchange.com/q/46097/44644. I'm new to information theory (and other relevant fields). It's even ...

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268 views

### Do all unitary representations weakly converge to zero at infinity?

Question. Let $G$ be a non-compact, finite dimensional Lie group, and let $(X, \mu)$ be a Radon measure space. Let $$\rho\colon G\to U(L^2(X))$$
be a unitary, strongly continuous, representation. Is ...

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497 views

### Almost all non-negative real numbers have only finitely many multiple lies in a measurable set with finite measure

I do not know whether this is the right place for posting this problem. But for several months I have no solution to this problem.
Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that ...

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43 views

### Minimizing the rate of geometric ergodicity of a Metropolis-Hastings kernel depending on a parameter

Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$.
I want to ...

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66 views

### Invariant ergodic measure Volterra operator on Continuous Functions

This is a follow-up to this question.
Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int_0^{\sqrt{\cdot}} f(s)ds.
$$
Is there an example of an ...

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63 views

### Invariant ergodic measure Volterra operator

Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int_0^{\cdot} f(s)ds.
$$
Is there an example of an ergodic and $V$-invariant Borel probability ...

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107 views

### The currents homology of closed orientable surfaces and Birkhoff Ergodic theorem?

I just know very little about currents but I need vexedly. Thanks for your help.
Let $M$ be a closed orientable surface and $I=(f_t)_{t\in[0,1]}$ be an isotopies from identity to $f$. Suppose that $\...

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257 views

### Invariant measure for composition on space of continuous functions

Let $C_g:C(\mathbb{R}^d;\mathbb{R}^d)\rightarrow C(\mathbb{R}^d;\mathbb{R}^d)$ be defined by $C_g(f)\triangleq f\circ g$ for some fixed $g \in C(\mathbb{R}^d;\mathbb{R}^d)$.
What are examples of ...

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158 views

### QI but not ME : not finitely presented groups!

I would like to know the examples of two groups which are not finitely presented and are quasi-isometric (QI), but they are not measured equivalent (ME) (in the sense of Gromov).
In the literature, ...

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89 views

### Is there a research direction within dynamical systems theory / ergodic theory that concerns conjugability to a two-point motion?

Let $X$ be a set equipped with some structure (e.g. topological space, measurable space, probability space, etc.). We say that two endomorphisms $f,g \colon X \to X$ are conjugate to each other if ...