All Questions
20 questions
0
votes
1
answer
169
views
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
5
votes
0
answers
160
views
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
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
3
votes
2
answers
250
views
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
views
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
views
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
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
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
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
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
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
1
answer
122
views
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
12
votes
1
answer
1k
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^...
3
votes
0
answers
227
views
Is there a noncommutative version of von Neumann's ergodic theorem? [closed]
The two most celebrated ergodic theorems are Birkhoff's ergodic theorem and von Neumann's ergodic theorem.
E. C. Lance in his remarkable work (Ergodic Theorems for Convex Sets and Operator Algebras) ...
- 3,486
1
vote
1
answer
92
views
Limit of doubly indexed functions
Let $(\Omega,\mu)$ be a $\sigma$-finite measure space and $f_{n,j}$ be a doubly indexed sequence of positive functions in $L^p(\Omega),$ $1<p<\infty.$ Suppose $f_{n,j}$ converges pointwise a.e. ...
- 128k
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 ...
CommunityBot
- 1
3
votes
1
answer
463
views
"Strongly mutually singular" families of measures, and the set of ergodic measures
Let $(X,\Sigma)$ be a measurable space [which we can assume to be a standard Borel space if we wish].
Let $\mathcal{S}$ be a set of probability measures on $(X,\Sigma)$. [If we wish, we can assume ...
- 6,206
3
votes
3
answers
370
views
How to show that there's a continuous function separating convex sets of Radon measures?
First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...
- 11
29
votes
1
answer
4k
views
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...
- 929