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This relatively straight forward. The main observation is that, in the Host Meiri terminology, the $p^{N}$-cells are exactly the inverse image of the $p^{N-1}$-cells, and the $\times p$ map is $p$ to …

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 …

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 …

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 …

Ok, let take $v$ in the bundle at $x$. We may decompose $v=v^{u}+v^{s}\in E^{u}\oplus E^{s}$. Assume without loss of generality that $\lVert v\rVert=1$. Applying $A^{i}$, using equivariance and Osceld …

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 …

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 …

Well, because this thread resurrected, I think I should mention the recent work by Bader and Furman generalizing Margulis' superrigidity theorem (and one might say simplifying it). In their work, the …

$\DeclareMathOperator\supp{supp}$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 …

[I'm assuming G means K] In case the elements of K are not measure preserving, I doubt that the question is correct. In general, one can easily show that entropy decrease by moving into factors. Her …

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.

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 …

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 …

EDIT - The answer below deals with an ergodic m.p.s As this question got up-voted, I've decided to fuly write a solution, based on the sketch I've made in the comments. Fix some $\varepsilon>0$ smal …

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 …