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$z \mapsto z^2$ is conjugated to $x \mapsto 2x$ mod 1 on ${\bf R}/{\bf Z}$. This map is in turn semiconjugated to the shift map $\sigma(\{a_n\}_{n\in {\bf N}} = \{a_{n+1}\}_{n\in {\bf N}}$ on $\{0,1\} …

The Jewett-Krieger theorem is done in the book of Petersen, Ergodic theory, section 4.4. It relies on Hindman's theorem, the proof is pretty long though (20 pages long). It is elementary in the sense …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

$A_{N,M}(f)$ 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 …

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 …

Mixing may be sufficient (not sure). Recall that a transformation is mixing if $\mu(T^{-k}A \cap B) \rightarrow \mu(A)\mu(B)$ as $k \rightarrow \infty $ for all $A$, $B$. If $T$ is just weak-mixin …

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 …

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 …