All Questions
Tagged with amenability ergodic-theory
11 questions
1
vote
1
answer
156
views
Nonamenable p.m.p. action on a standard probability space
Let $G$ be a discrete nonamenable countable group acting on a standard probability space $(X,\mu)$ through measure-preserving transformations.
Is the action of $G$ always amenable?
(Amenable action, ...
- 113
7
votes
1
answer
158
views
Density of “diagonal sets” in amenable groups
Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \...
- 323
0
votes
2
answers
224
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 $|Gx| = n$...
- 19.7k
2
votes
0
answers
99
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 ...
- 4,432
2
votes
0
answers
109
views
Non-ergodic Dye Theorem for orbit equivalent automorphisms
The Dye Theorem states that any two free ergodic p.m.p automorphisms of a standard probability space are orbit-equivalent.
Question: Is there a version of the above theorem for non-ergodic ...
- 3,897
5
votes
1
answer
630
views
Characterizing residually amenable groups
Let $G$ be a finitely generated group. The amenability of $G$ is equivalent to the existence of a certain "weak measure" on $G$. Is there such a characterization for residually amenable groups as well?...
- 11.3k
3
votes
1
answer
359
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Pointwise ergodic theorem for amenable semigroups
Using tempered Følner sequences one may show a pointwise ergodic theorem for amenable groups.
(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)
Is there a ...
- 123
7
votes
1
answer
330
views
Do syndetic sets on amenable semigroups have positive upper density?
Let $\mathbb{G}$ be a discrete amenable semigroup, and $\left\{ F_{n}\right\} $
a Folner sequence.
For $S\subset \mathbb{G}$ define the upper density as $D^{\ast
}(S)=\limsup_{n\rightarrow \infty }\...
- 123
11
votes
2
answers
3k
views
Characterization of amenable actions
Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every ...
- 113
5
votes
0
answers
245
views
Can invariant means be really considered as the generalization of the uniform measure?
I am writing a paper for game theorists where I use (countable) amenable groups to do some things. So I am writing up a preliminary section about countable amenable groups whose main purpose is to ...
- 6,034
15
votes
3
answers
2k
views
Alternative proofs of the Krylov-Bogolioubov theorem
The Krylov-Bogolioubov theorem is a fundamental result in the ergodic theory of dynamical systems which is typically stated as follows: if $T$ is a continuous transformation of a nonempty compact ...
- 6,206