All Questions
50 questions
4
votes
1
answer
130
views
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 ...
- 257
7
votes
1
answer
211
views
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 \...
- 113
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};\...
- 10.6k
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
2
votes
0
answers
126
views
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
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 ...
- 59
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
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 $...
- 1,658
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 ...
- 307
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 \...
- 6,205
4
votes
1
answer
446
views
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 ...
- 333
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 ...
- 163
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 ...
- 73
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
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+\...
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 $\mu$ ...
- 51
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
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?
- 1,879
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 ...
0
votes
1
answer
409
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<\...
- 1,879
0
votes
0
answers
369
views
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 ...
- 1,879
3
votes
1
answer
233
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:\...
- 167
3
votes
1
answer
372
views
Attractors in random dynamics
Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
- 333
2
votes
0
answers
152
views
Baker map-like 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 \...
- 2,069
2
votes
0
answers
139
views
size of local strong stable manifold is measurable
Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...
- 199
2
votes
1
answer
119
views
time delay ergodic theorem
given dynamic system $(X, \mathcal{B}, F, \mu), \mu \circ F^{-1}=\mu, F $ is mixing, $ A \in \mathcal{B}, s.t. \mu(A) >0 $.
consider dynamic system $(X\times X, \mathcal{B}\otimes \mathcal{B}, ...
- 553
13
votes
1
answer
559
views
Entropy of composition
I asked this at math.stackexchange.com, but got no answers.
Let $(X,B,\mu)$ be a probability space. Let $T,S:X→X$ be two measurable measure preserving maps that commute (i.e $TS=ST$). Let $A$ be a (...
- 133
4
votes
1
answer
181
views
What are invariant measures of $E_m \times R_\alpha$ on $S^1 \times S^1$? Are they necessarily product measures?
For $m \in \mathbb{N}$, let $E_m \colon S^1 \to S^1$ be multiplication map $x \mapsto mx$. Also, let $R_\alpha$ be the map $x \mapsto x+\alpha$.
Now, consider $E_m \times R_\alpha \colon S^1 \times S^...
- 43
15
votes
0
answers
3k
views
Weak$^*$ convergence of measures vs. convergence of supports
Let $X$ be a compact metric space and let $\mathcal M(X)$ denote the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\text{supp} \mu$ for the support of $\mu$. It is easy to ...
- 1,658
6
votes
1
answer
469
views
Poincare Recurrence by Mean Ergodic Theorem
I have a question regarding a confusion from reading the Princeton Companion to Mathematics on the topic of Ergodics Theorems. It is about proving a stronger version of Poincare Recurrence Theorem ...
- 1,245
2
votes
1
answer
200
views
Measurable isomorphism between two non-totally ergodic systems
Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
- 319
1
vote
0
answers
197
views
A certain measure on Banach algebras
According to the comments of Nate Eldredge I did revise the question. In particular I change "$C^{*}$ algebras" to "Banach algebras".
Is there a reference who introduce the following measure on ...
- 356
6
votes
3
answers
484
views
A question on invariant measures
Let $(X, \mathcal{B}, T)$ be a topological dynamical system and $M(X, T)$ be the set of all invariant measures.
I do not know is there some nice functional characterization of the following set
$\{...
- 1,706
3
votes
2
answers
340
views
Convex combinations of Bernoulli Measures
How big is the weak-* closure of the set of all (finite) convex combinations of Bernoulli measures among all invariant probability measures?
I mean, we are in the symbolic space $\{1,2,\ldots,d\}^{\...
4
votes
1
answer
211
views
Is there a mixing condition to get the decay property I want?
Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation.
Consider a measurable set $A\subset X$ with $0<\mu(A)<1$
For each $N$ define ...
- 2,964
7
votes
0
answers
305
views
Generator of a $\bigoplus_{n=0}^\infty \mathbb{Z}/2\mathbb{Z}$-action
Let $T$ be a measure-preserving action of a group $G$ on a Lebesgue space $X$. That means that $T$ associates an automorphism (i.e. an invertible measure-preserving transformation) $T^g$ of $X$ to ...
- 2,560
5
votes
1
answer
153
views
Generator determined by finitely many translates implies zero entropy
Let $T$ be a measure preserving transformation of a standard probability space $(X,\mathcal{B},\mu)$. A partition $\alpha$ of $X$ is said to be a generator for $T$ if the smallest $T$ invariant $\...
- 1,769
7
votes
2
answers
409
views
List of Bernoulli chaotic systems
Which discrete chaotic systems are known to be Bernoulli (i.e. measure theoretically isomorphic to a Bernoulli shift, one-sided or two-sided)?
I am aware that it is known for some uniformly ...
- 225
0
votes
0
answers
182
views
On a certain set of probability measures on a shift
Denote by $\mathbb{Z}_2=\{0,1\}$ the integers modulo 2.
Let $S:\mathbb{Z}_{2}^{\mathbb{N}}\times\mathbb{Z}_{2}^{\mathbb{N}} \rightarrow \mathbb{Z}_{2}^{\mathbb{N}}$ be the sum $S(a,b) = a+b$, where $...
1
vote
1
answer
174
views
Is it possible to define the density of the logistic map for $x<0$?
Probability density functions (PDF's) have inherent connections to the field of
Dynamical Systems.
The motivation for this question can be found in: http://www.stat.cmu.edu/~cshalizi/754/2006/notes/...
- 1,197
2
votes
1
answer
440
views
Weak Convergence to Lebesgue Measure
I'm trying to understand the proof given by D. Rudolph in his paper "x2 and x3 invariant measures and entropy". I'm particularly trying to undestand the proof of lema 4.4.
Let's consider a secuence ...
- 123
1
vote
0
answers
106
views
A argument related measurable partitions in dynamic system
$X$ is a compact metric space, and $T:X\rightarrow X$ be a continuous map, which is finite to one. Denoted by$ X_{0}$ the set of all points $x\in X$, such that for all sufficiently small $\epsilon>...
- 1,706
3
votes
1
answer
1k
views
Liouville's theorem: How to get an invariant measure?
Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows
\begin{equation}
d\mu = \frac{d\sigma}{|| ...
- 139
4
votes
2
answers
231
views
The relations between conservative part and conservativity
I revised the question. In smooth ergodic theory, a diffeomorphism is said to be conservative (I), if it preserves the Lebesgue measure. So for some of us, conservativity is just short for measure-...
- 2,244
9
votes
2
answers
1k
views
Fourier transform of x2 invariant measure
Let $T:\mathbb{R}/\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}$ be the map defined by $T(x)=2x$, and suppose that $\mu$ is a $T$ invariant and ergodic Borel probability measure on the space, which is ...
- 1,723
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 ...
- 897
2
votes
2
answers
557
views
trivial map on $\sigma-$algebra $\mod{}0$ is trivial
Hi everyone!
I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
- 23
2
votes
1
answer
1k
views
Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?
It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's ...