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Theorem 2.1 states that the set of ergodic measures is a $G_\delta$ set in the set of invariant probability measures. This means that this set is a countable intersection of open sets, this does not i …

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 …

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 …

Since (p_1....,p_n) is an eigenvector of the transition matrix, your uniformity assumption is satisfied iff the transition matrix associated to the partition is bistochastic. I guess this is rarely th …

The mean ergodic theorem on L^p spaces is due to F. Riesz (1938) and S. Kakutani (1938). For p in $[1,\infty[$, this is theorem 1.2 ff in the book of Krengel, "Ergodic theorems". If $p=\infty$, you g …

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 …

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 …

Note that the flow must not have fixed points. Global sections allow to reduce the study of a flow to the study of a transformation. This is often the reason invoked in differential dynamics in order …

An Anosov flow has many periodic orbits. Take the mean of two Dirac measures on two distinct periodic orbits and you get an invariant probability measure that is not ergodic. The standard proof given …

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) …

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 …

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 …

Here is another way to find a fundamental domain. First identify $X$ with $[0,1]$. You want to pick a single point in each orbit of the action. Just take the smallest one. Let be more specific. Cons …

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 …

Axiom A diffeomorphisms have this property. The following result is due to Bowen and Ruelle. Theorem Let $X$ be a connected compact manifold and $T : X \rightarrow X$ be an Axiom A $C^2$ diffeomorphis …