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Results tagged with ergodic-theory
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user 8857
Dynamical systems on measure spaces, invariant measures, ergodic averages, mixing properties.
11
votes
Accepted
Furstenberg $\times 2 \times 3$ conjecture, bibliography
Well that will be some lengthy answer.
The first article that was published after the famous disjointness paper is another paper by Hillel called "Intersections of Cantor sets", it's related to the m …
10
votes
Accepted
Equidistibution of horocycles through Hecke eigenvalues of Maass cusp forms
There are two ways to solve this problem - one by ergodic methods, and the other one using purely harmonic methods.
The harmonic method you are indicating is just to take the delta function of the po …
8
votes
Has dynamics on $G/\Gamma$ ever been used to prove interesting things about $\Gamma$?
There's a nice proof by Margulis showing that arithmetic subgroups are indeed lattices using the famous Dani-Margulis non-divergence theorem.
Actually if you will investigate Ratner's original formula …
6
votes
Examples of transformations that are totally ergodic but not weakly mixing?
Totally ergodic is equivalent to not having rational eigenvalues (I guess a suitable reference for this is Eli's book).
Hence basically the Kronecker factor of such a system will be "essentially" the …
5
votes
Book recommendation for ergodic theory and/or topological dynamics?
I second Siming Tu's recommendation for E-W book.
It is a well balanced book (regarding theory vs applications), it has nice appendix contains relevant theory from functional analysis, and it contains …
5
votes
Uniform distribution of sequence mod 1
It is unclear what is "most $r$'s even mean.
A standard argument would show that for any increasing sequence, for Lebesgue almost every $x$, $a_{n}.x$ is equidistributed mod $1$.
For the case of power …
4
votes
Hausdorff dimension of sequence space
This observation is attributed to H. Furstenberg, and appears (in the case of shift-invariant sets, i.e. Cantor sets) in his beautiful Disjointness paper (in section $3$, which you can read independen …
4
votes
"Typical" convergence rate for the von Neumann mean ergodic theorem
In general, it varies.
There are cases where the convergence is quite fast (for example in the case where the system is mixing, and say in the presence of spectral gap, think of Bernoulli system or sa …
4
votes
Looking for at least one beautiful and not too technical result in asymptotic group theory
You can speak about the Howe-Moore theorem, which is very useful and imply ergodicity (and actually, mixing) of group actions on reasonable spaces.
3
votes
Accepted
Uniquely ergodicity and polynomial ergodic average
This is indeed true for some "nice systems", for example one can show this theorem (for say $L^{2}$-functions) for Kronecker systems simply by van-der-Corput trick.
In general, those averages converg …
3
votes
Accepted
Furstenberg-Zimmer theorem: non-invertible systems
Posted as requested - consult the book by Manfred Einsiedler and Tom Ward - "Ergodic Theory with a view towards number theory" - published in GTM, especially in ch 7.
3
votes
Diophantine equations and ergodic theorems
It is a whole line of ideas (and proofs) which go usually by the name of ``Linnik's problems''.
Apart from Linnik's book (and the Linnik-Skubenko theorem), it has been extensively studied by many rese …
2
votes
Accepted
Continuity of relative entropy with respect to the weak* topology
Now I'll post some sort of an answer, and not just a comment due to the length.
Your first question is answered above.
For the second one, if you're willing to take the minus inside (take the inverse …
2
votes
Ratner's orbit closure for a unipotent semigroup
$\DeclareMathOperator\supp{supp}$The theorem holds for semigroups as well (well, in the finite volume setting! in the infinite volume setting there are subtleties between two-sided and one-sided avera …
2
votes
How to show the geodesic orbit of a badly approximable number are/are not homogeneously equi...
A number is in BA if its orbit is bounded. Any such orbit closure must contain a full $A=\langle g_t\rangle$ orbit. By examining the possible subgroups, any such hypothetical $H$, as a stability group …