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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 …

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) …

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 …

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) …

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 …

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 …

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 …

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 …