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Results tagged with ergodic-theory
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user 8176
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
4
votes
pointwise ergodic theorem and mean sojourn time
I can show that
$$\int_X f(x) \ d \mu= \liminf_{k \to \infty} \frac{1}{|F_k|} \sum_{g \in F_k} f(g.x)$$ for $f \in L^1(X)$ with $f \geq 0$ and almost all $x \in X$.
The conclusion is obvious if $f \i …
- 25.5k
7
votes
Accepted
amenable equivalence relation generated by an action of a non-amenable group
The answer is yes, such an action exists.
What is needed for the construction is the following very nice example of an action of a non-amenable group on $\mathbb Z$, which I just learned from Gabor E …
- 25.5k
21
votes
Accepted
Central Limit Theorem(s) for irrational rotation
The result depends on the approximation properties of $\alpha$.
Of course one has to assume $\int_{S^1} f(z)dz=0$. A rotation by $\alpha$ has the effect that the $k$-th Fourier coefficient of $f$ is …
- 25.5k
5
votes
Accepted
Characterizing residually amenable groups
It is easy to see that a finitely generated group is residually amenable if and only if there exists an bi-invariant ultra-metric on $G$ and a finitely additive $G$-invariant measure on open (with res …
- 25.5k