All Questions
16 questions
2
votes
0
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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
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
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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
0
votes
0
answers
118
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A measure on the group of homeomorphisms of $\mathbb T^2$
Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost
everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
- 101
1
vote
0
answers
177
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Building random homeomorphisms of the torus $\mathbb T^2$
In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
- 101
1
vote
0
answers
96
views
Building random homeomorphisms of the circle
Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as
\...
- 101
3
votes
0
answers
115
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Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
- 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
3
votes
1
answer
131
views
Questions about ratio set in a dynamical system
Given a dynamical system $(X, \Omega, \mu)$ ($\Omega$ the $\sigma$-algebra and $\mu$ a measure), we assume there is a group $G$ acting on the system in the sense that, for each group element $g$, $g$ ...
- 307
1
vote
1
answer
238
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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
3
votes
1
answer
361
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Equivalent definitions of strongly proximal action
Consider the following fragment from the paper "C*-simplicity and the unique trace property for discrete groups" by Breuillard, Kalantar,
Kennedy and Ozawa:
I have two questions:
(1) What ...
- 175
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
279
views
Set operations over iterated function systems
An iterated function system (IFS) is a finite set of contraction mappings on a complete metric space. Symbolically,
$$\{f_i :X \to X \mid i=1,2,..n \},\quad n \in \mathbb{N}$$
is an IFS if each $...
- 21
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
12
votes
1
answer
838
views
A measure theory question
Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...
- 1,143
2
votes
1
answer
543
views
Approximation of the radon-derivative
I am looking for the following statement.
Let $X$ be a topological space and let $\mu$, $\nu$ be Radon-Borel measures in the same measure class i.e. the the zero-sets are identical. Denote then by $...
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