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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 …
Asaf's user avatar
  • 2,459
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 …
Asaf's user avatar
  • 2,459
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 …
Asaf's user avatar
  • 2,459
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 …
Asaf's user avatar
  • 2,459
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 …
Asaf's user avatar
  • 2,459
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 …
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  • 2,459
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.
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  • 2,459
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 …
Asaf's user avatar
  • 2,459
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 …
Asaf's user avatar
  • 2,459
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 …
Asaf's user avatar
  • 2,459
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 …
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