All Questions
Tagged with measure-theory ergodic-theory
125 questions
3
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0
answers
61
views
On the relative growth rates of occupancy times in ergodic theory
Let $(X, \mathcal{F}, \mu)$ be a general measure space, and let $T: X \to X$ be a measure-preserving transformation on $X$. Assume that $T$ is ergodic and satisfies the property that, for any set $A \...
- 105
7
votes
1
answer
211
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Existence of asymptotic sequence in ergodic measure-preserving transformations
Let $(X,\mathcal{F},\mu)$ be a measure space and let $T:X\to X$ be an ergodic measure-preserving transformation. We assume that $T$ satisfies the property that if $B \in \mathcal{F}$ and $T^{-1}B \...
- 63.9k
0
votes
1
answer
169
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Existence of a "universal" measure-preserving transformation on the unit interval
Let $I = [0,1]$ be the unit interval equipped with the Lebesgue measure $\lambda$. Let $\mathcal{M}$ be the set of all Lebesgue measure-preserving transformations $T: I \to I$. We say a transformation ...
- 23.2k
4
votes
1
answer
130
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Restrict sigma algebra in measure-preserving system
Consider a measure space $(X,\mathcal{A},\mu)$ and a measure-preserving transformation $\phi \colon X\rightarrow X$, that is, $\phi$ is measurable and $\phi_*\mu = \mu$.
My intuition tells me that we ...
- 12.9k
3
votes
1
answer
130
views
Do sets of big returns contain sets of returns?
We say a subset $E$ of $\mathbb{N}$ is a set of returns if there is some measure preserving system $(X,\mathcal{B},\mu,T)$ and some $A\in\mathcal{B}$ with $\mu(A)>0$ such that $E=\{n\in\mathbb{N};\...
- 929
5
votes
0
answers
160
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Hartman uniform distribution of means
Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
2
votes
0
answers
92
views
Existence of ergodic subgroup invariant to a product measure
Let $X=\{0, 1\}^{\mathbb{N}}$ and $G$ be the group of permutations, each of which only permutes finitely many coordinates of $X$. Fix a sequence $(\lambda_n)_{n\in \mathbb{N}} \subseteq (0, 1]$ and ...
- 307
4
votes
1
answer
446
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Birkhoff ergodic theorem for ergodic Markov processes
This question was previously posted on MSE.
This question might be easy but I am really stuck on it.
Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
CommunityBot
- 1
2
votes
0
answers
126
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Identification of Maharam extension
All definitions used in this post are from Björklund, Kosloff, and Vaes - Ergodicity and type of nonsingular Bernoulli actions. This post is inspired by the beginning of Section 2.2 in the same paper, ...
- 307
2
votes
1
answer
206
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Deriving an inequality for the integral of maximum indicator functions under measure-preserving transformations
Let's denote the measure space by $(X, \mathcal{B}, \mu)$ and the measure-preserving transformation by $T: X \to X$. Let $A \in \mathcal{B}$ be a measurable set with $0 < \mu(A) < \infty$. Let $...
- 578
1
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1
answer
114
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Ensuring the measure condition $\mu(E) = \lambda$ in a lemma: need some clarification regarding the selection of $A$
I was studying a lemma from my notes on ergodic theory and encountered a difficulty. The lemma states:
Let $(X, \mathcal{B}, \mu)$ be an infinite non-atomic measure space, and let $T$ be an ergodic ...
- 111
3
votes
2
answers
250
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Existence of a positive measurable set with disjoint preimage under iterated transformation
Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal ...
- 10.6k
2
votes
2
answers
328
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Existence of the limit of periodic measures
Let $T: X \to X$ be a continuous map over a compact metric space. We say that a measure $\mu$ is $T$-invariant if $T_{\ast} \mu= \mu$. We denote by $M(X, T)$ the space of all $T$-invariant Borel ...
- 1,043
15
votes
1
answer
1k
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Krein Milman theorem without the axiom of choice
The Krein-Milman theorem asserts that in a locally convex topological vector space, a nonvoid compact convex subset is the closed convex envelope of its extreme points. But I would like to know when ...
- 12.9k
12
votes
3
answers
891
views
Looking for at least one beautiful and not too technical result in asymptotic group theory
We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
- 1,446
0
votes
1
answer
282
views
Do invariant open sets generate the $\sigma$-algebra of invariant sets?
Let $X$ be a Polish space with Borel $\sigma$-algebra $B(X)$. Let $G$ be a locally compact group. $T:G\times X\to X$ be a continuous action of $G$ on $X$.
The $\sigma$-algebra of invariant sets is ...
- 6,406
1
vote
0
answers
134
views
Mathematical justification for the use of an energy shell in the microcanonical ensemble
I would like to understand an identity used in the deduction of the explicit formula for the probability distribution of the microcanonical ensemble in statistical mechanics.
Consider $\Lambda$ to be ...
- 1,305
1
vote
0
answers
94
views
Convex combination of positive mean-ergodic operators
Let $T_1,T_2:L^1([0,1],\mathrm{d}x)\to L^1([0,1],\mathrm{d}x)$ be positive mean-ergodic operators such that:
For every $h:[0,1]\to \mathbb{R}_+$ we have that
$$\int_0^1 T_1 h(x)\mathrm{d}x = \int_0^1 ...
- 333
0
votes
0
answers
101
views
Approximation arguments
I am a student reading about Luo and Sarnak's paper and I have trouble understanding the conclusion.
In the paper this theorem is proved for a continuous function of compact support $\psi$:
$$\...
- 869
25
votes
6
answers
6k
views
Proof of Krylov-Bogoliubov theorem
Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
- 4,713
3
votes
0
answers
160
views
Ergodic diffeomorphisms of the circle
From the paper
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a ...
- 101
2
votes
0
answers
163
views
Weakly mixing diffeomorphism
From
Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889.
the following result is known: Let $(E,\Sigma, \mu)$ be a measure ...
- 101
7
votes
1
answer
253
views
Are all quasi-regular points on Polish spaces generic points?
Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
- 1,074
2
votes
1
answer
166
views
In general is $\frac{d\,\mu_1}{d\,\mu_2}\circ T = \frac{d\,T\mu_1}{d\,T\mu_2}$?
Given an ergodic and non-singular dynamic system (definition provided here) $(X, \mathcal{B}, \mu_1, T)$ where $(X, \mathcal{B}, \mu_1)$ is a measure space and $T$ is a fixed transformation, we then ...
- 1,095
1
vote
1
answer
182
views
Friedland metric entropy
I was asking if it is possible to extend the definition of topological Friedland entropy for $\mathbb{Z}^d$ continuos actions to measure preserving actions.
The topologica Friedland entropy is ...
CommunityBot
- 1
2
votes
2
answers
286
views
Measure preserving transformation that makes two partitions independent
I am looking for a reference for the following result. I think it is well known but I haven't seen it written down anywhere.
Let $(X, \mathcal{B}, \mu)$ be a standard measure space and let $\mathcal{...
- 1,769
2
votes
0
answers
124
views
Almost periodic functions in weak mixing extension
In Theorem 3.4.6 of the 'Lecture notes on ergodic theory' by Jesse Peterson, it is shown that in a weak mixing extension, every almost periodic function is trivial. I have a doubt in the proof of this ...
- 85
3
votes
0
answers
217
views
The baker problem
Let $S =[0, 1]^2$ denote the unit square in $\mathbb R^{2}$. For any subset $A$ of $S$ let $A^{c}$ denote its complement in $S$, and $\overline{A}$ its closure in $S$. Given a measurable map $g: W \...
- 6,155
2
votes
0
answers
123
views
Probability of a finite cylinder set in a free group
Let $\mathbb{F}_n$ be the free group (each elemen is in its reduced form) generated by the set $\Sigma_n = \{a_1, a_2, \cdots, a_n, a_1^{-1}, a_2^{-1}, \cdots, a_n^{-1}\}$ and let $e$ denote the ...
- 307
2
votes
1
answer
131
views
Absolute continuity of geometric sum of ergodic random variables
I have a stationary ergodic process of random variables $\{X_n\}_{n\in\mathbb{N}}$ and a constant $r<1$. Suppose that $\mathbb{E}(\log^+|X_1|)<\infty$. Then the sum
$$
\sum_{n=1}^{\infty} r^nX_n ...
- 50.6k
9
votes
1
answer
950
views
Sort-of converse of Kolmogorov zero-one theorem
Let $(\Omega, \mathscr F, \mathbb P)$ be a probability space. The Kolmogorov zero-one theorem states that
Suppose we have independent random variables $X_1, X_2, ...$. Then $\forall \ A \in \bigcap_n ...
- 63.9k
6
votes
1
answer
205
views
Existence of a continuous ergodic dynamical system for a given distribution?
It seems to me that given a distribution (which is well-behaved), there should be at least an ergodic dynamical system that its time average would create this distribution. Is this question already ...
- 6,571
1
vote
1
answer
238
views
The mean ergodic theorem for weakly mixing extension
I asked this question in https://math.stackexchange.com/q/4236870/528430, but did not get any help.
I got stuck with the following while going through the proof of Lemma 3.21 from the book 'Ergodic ...
- 63.9k
0
votes
0
answers
153
views
Ergodic action on product spaces
Let $(X_1 \times X_2,d\mu)$ be a measure space with $X_2$ compact. Suppose that we have a continuous (diagonal) action of a topological group $G$ on $X=X_1 \times X_2$. I know that the action of $G$ ...
- 59
2
votes
1
answer
204
views
Exponential mixing for subshifts
I asked this question on Math.StackExchange some time ago and got no responses.
Let $G=(V,E)$ be a finite graph with adjacency matrix $A$. Let us consider the associated subshift of finite type
$$
\...
- 2,621
8
votes
1
answer
647
views
How to analytically prove chaos
Consider the following map
\begin{align*}
T \colon \mathbb{R}\times\mathbb{S}^1 \to & \mathbb{R}\times\mathbb{S}^1 \\
(x,\theta) \mapsto & \left(\frac{x}{4}+ \sin^2\left(\pi\left(\theta+\...
- 177
1
vote
0
answers
156
views
Quotient measure on locally compact spaces
Suppose we are given a locally compact topological space $X$ and a discreet group $G$ acting on it (we can assume the action to be proper). Given a Radon probability measure on the quotient space $G \...
- 59
1
vote
1
answer
137
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Ergodic theorem on limit of periodic transformations?
Suppose $(X,\mu)$ is a probability space, and $T_n, n \in \mathbb N$, is a sequence of periodic measure preserving transformations. For $x \in X$ and $f : X \to \mathbb R$, let $\mathrm{avg}_{f,n}(x)$...
- 23.2k
2
votes
2
answers
209
views
Does ergodic theorem apply to trajectories outside of attractor?
Ergodic theorem says that $\displaystyle\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{t=1}^nf(T^tx) = \displaystyle\int f\,\mathrm{d}\mu$ for $\mu$-almost every $x$. In many cases, the support of $\mu$ ...
- 3,928
2
votes
0
answers
166
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Two generalizations of the Verblunsky Theorem
I learned from this paper about the Verblunsky theorem.
My question is that: What kind of generalizations of this theorem is availlable?
In particular I am interested in the following two possible ...
- 356
14
votes
1
answer
655
views
Almost all non-negative real numbers have only finitely many multiples lying in a measurable set with finite measure
Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows:
$$S:=\{t\in [0,\infty):nt\...
- 24.2k
1
vote
0
answers
77
views
Is there a term for a linear operator on an $L^p$ space that "locally respects boundedness"?
Let $X$ be a Polish space, and $\mu$ a locally finite measure. Take any $p \in \{0\} \cup [1,\infty)$. We will say that a linear operator $T \colon L^p(\mu) \to L^p(\mu)$ has property $(\ast)$ if ...
- 708
0
votes
1
answer
88
views
Recurrence results for an "on average" measure preserving transformation
I have a finite measure space $(X, \mathcal{S}, \mu)$, and a transformation $f:X\rightarrow X$ that "preserves measure on average". That is, for $A \in \mathcal{S}$
$$
\lim_{n\rightarrow \...
- 1,087
2
votes
2
answers
212
views
Is Birkhoff's ergodic theorem true for $L_\infty$?
Is Birkhoff's pointwise/individual ergodic theorem for $L_\infty.$ Clearly, it is true if the measure space is finite? What about the measure space not finite?
- 23.2k
0
votes
2
answers
221
views
Induced probability measure on a finite orbit under a group action
Suppose we have a discrete group $G$ acting on a compact set $X \subseteq \mathbb{R}^d$
via measure-preserving homeomorphisms, and suppose we have a point
$x$ whose orbit $Gx$ is finite (say $|Gx| = n$...
- 17k
1
vote
1
answer
191
views
What are the hypotheses we should add for the generalizations of Furstenberg recurrence theorem?
In my question here I suggest a possibility for generalization of Furstenberg recurrence theorem needing some hypothesis for that generalization to be hold in the side of convergence of the below ...
- 14.2k
2
votes
1
answer
159
views
Can we show that this transition semigroup preserves a certain Wasserstein space?
Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...
- 167
1
vote
1
answer
189
views
If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure?
Let $E$ be a separable $\mathbb R$-Banach space, $\rho$ be a complete separable metric on $E$, $\operatorname W_\rho$ denote the Wasserstein metric of order $1$ associated to $\rho$, $\mathcal M_1(E)$ ...
- 29.7k
1
vote
1
answer
183
views
If $(κ_t)$ is a semigroup with invariant measure $\mu$ and $ν$ is singular to $\mu$, then $νκ_t$ might not converge to $\mu$ in total variation norm
Let $E$ be a Polish space, $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$, $\mu$ be a probability measure on $(E,\mathcal B(E))$ invariant with respect to $(\kappa_t)_{t\ge0}$ and $\...
- 17k
2
votes
1
answer
122
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How is this bound for a Wasserstein contraction coefficient in this paper obtained?
I'm trying to understand the following conclusion from this paper (see below for the relevant paragraphs):
I'm not sure whether they really mean that it follows from the statements of Lemma 3.2 (...
- 230