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Results tagged with ergodic-theory
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user 10457
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
2
votes
Quantitative versions of ergodic theorem
A. Leibman proved a quantitative lower bound for the averages $\frac{1}{N}\sum_{n=0}^{N-1} \mu(A\cap T^{-n}A)$ in terms of $\mu(A)$ (note: the sum begins at $n=0$). The bound is
$$
\frac{1}{N}\sum_{n …
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5
votes
Uniquely ergodicity and polynomial ergodic average
Theorems 1.2 and 1.3 of R. Pavlov's article Some counterexamples in topological dynamics provide strong counterexamples for the convergence of $\frac{1}{N}\sum_{n=1}^{N}f(T^{n^2}x)$; in particular the …
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3
votes
Extension of Khintchine's recurrence in a simple case
(Edit: my original remark didn't take all of the question into account. One can say that the Kronecker factor does retain sufficient detail to prove the desired result.)
The large intersection proper …
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4
votes
Accepted
Ergodic decomposition of the action of a subgroup
The answer to both questions is "no" - a counterexample (labeled "folklore") appears immediately prior to Question 6.6 in Austin, Tim, Extensions of probability-preserving systems by measurably-varyin …
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2
votes
Accepted
Transformation extending all ergodic rotations
There is an ergodic system $\mathbf Z = (Z,m,R)$, such that $\mathbf Z$ admits every irrational rotation as a factor.
As mentioned in the comments, such a system $\mathbf Z$ cannot have $L^2(Z,m)$ se …
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5
votes
Accepted
Do sets of big returns contain sets of returns?
No, a counterexample is due to Alan Forrest:
Forrest, A. H., The construction of a set of recurrence which is not a set of strong recurrence, Isr. J. Math. 76, No. 1-2, 215-228 (1991). ZBL0773.28014.
…
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21
votes
Accepted
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Manfred Einsiedler and Alexander Fish have a paper (arxiv.org/abs/0804.3586) showing that a multiplicative subsemigroup of $\mathbb N$ which is not too sparse satisfies the desired measure classificat …
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4
votes
Accepted
A property of rapid sequences of natural numbers
A sequence is called lacunary if, in your terminology, its minimum growth rate is strictly greater than $1$. The following articles prove that every lacunary sequence is remote. If I understand your …
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5
votes
0
answers
130
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 …
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