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I know of six proofs of the Birkhoff ergodic theorem. using a maximal inequality (Birkhoff, Riesz, Wiener, Yosida, Kakutani, Garsia...) based on martingales and upcrossing inequalities (Bishop 1966) …

The sequence $\sum a_n$ is unbounded. This is a consequence of a general result from Kesten, On a conjecture of Erdös and Szüsz related to uniform distribution mod 1, Acta Arithmetica (1966). The pr …

Topological mixing is called "permanent regional transitivity" (and demonstrated) by Gustav Hedlund in his 1939 article on the dynamics of the geodesic flow in constant negative curvature (third page) …

These are called good universal sets. Bourgain (1987) proved that sequences of the form $p(n)$, $n \in {\bf N}$, $p$ a non constant polynomial, are good. He also proved (1988) that the set of primes …

Yes it works. No, this can't be deduced from the Poincare recurrence theorem. If it was possible, the ergodic hypothesis would not be needed. But without the ergodic hypothesis, it is easy to give a …

Let $\Omega$ be the standard volume on your Riemannian manifold, and $\phi$ a smooth function on M. A quick computation shows that $e^\phi \Omega$ is invariant by f if and only if the following cohomo …

It depends on $U$ of course. If $U$ is the identity, the convergence is pretty fast. It also depends on $f$. If $f$ is a coboundary ($f=g-Ug$), then the convergence is of the order $1/n$. When $U$ co …

$A_{N,M}(f)$ converges to some $U$-invariant function $g$ that satisfies $\langle g, f\rangle$ = $\langle g, g\rangle$. We also have $\langle g, 1\rangle = \lim \langle A_{N,M}(f), 1\rangle = \langl …

Anosov flows are ergodic and a geodesic flow can be Anosov even if the curvature is not strictly negative. This was studied by Eberlein in the seventies, in an article from 1973 entitled when is a geo …

A colleague pointed out the following counterexample. Let $h_t$ be the horocyclic flow on a negatively curved compact surface S. This R action is known to be minimal. Now Consider the $R^2$ action on …

Answer to the quick version. Yes it is true as soon as $(X,\mu)$ is a Lebesgue space. Beware that the transformation on the product $A_i\times B_i$ is not necessarily a true product, but instead it is …

1 - There is an explicit reference given in the book: Borel, Harish-chandra, arithmetic subgroups of algebraic groups, 1962. This is the general result for matrix groups. A simpler proof has been giv …

This is an old question but I don't see the obvious answer, so here we go. A huge field of research in mathematical physics during the XIXe century revolved around giving explicit solutions to the eq …

Since the weak topology is Hausdorff, there is an open set containing $m$ and disjoint from $K$. So we can find $f_1,...,f_n$ and $\varepsilon$ such that the open set $$\{\mu \mid \int f_i dm -\vareps …

Weak mixing is generic. The result is due to K. R. Parthasarathy, "Indian Journal of Statistics", November 1962, Series A vol.24. Note that in the measurable setting, this is due to Halmos (see his 1 …