Glossary 1 1. Definition of the subject and its importance 2 2. Basic Results 2 3. Panorama of Examples 7 4. Mixing notions and multiple recurrence 10 5. Topological group Aut(X,μ) 13 6. Orbit theory 15 7. Smooth nonsingular transformations 21 8. Spectral theory for nonsingular systems 22 9. Entropy and other invariants 25 10. Nonsingular Joinings and Factors 27 11. Applications. Connections with other fields 31 12. Concluding remarks 36 References 36

pdf/ergodic-theory-nonsingular-transformations-4sla7gwtyl.pdf

Ergodic Theory: Nonsingular Transformations

For a Cantor set X, let Homeo(X) denote the group of all homeomorphisms of X. The main result of this note is the following theorem. Let T ∈ Homeo(X) be an aperiodic homeomorphism, let μ 1 ,μ 2 ,...,μ k be Borel probability measures on X, and let e > 0 and n > 2. Then there exists a clopen set E C X such that the sets E,TE....,T n-1 E are disjoint and μ i (E ∪ TE ∪... ∪ T n-1 E) > 1-e, i = 1,..., k. Several corollaries of this result are given. In particular, it is proved that for any aperiodic T ∈ Homeo(X) the set of all homeomorphisms conjugate to T is dense in the set of aperiodic homeomorphisins.

/pdf/the-rokhlin-lemma-for-homeomorphisms-of-a-cantor-set-27886mv3yw.pdf

The Rokhlin lemma for homeomorphisms of a Cantor set

We show that Chacon's non-singular type III _\lambda transformation T_\lambda , 0 , is power weakly mixing, i.e. for all sequences of non-zero integers \{k_{1},\dotsc,k_{r}\} , T_\lambda^{k_{1}}\times\dotsb\times T_\lambda^{k_{r}} is ergodic. We then show that in infinite measure, this condition is not implied by infinite ergodic index (having all finite Cartesian products ergodic), and that infinite ergodic index does not imply 2-recurrence.

/pdf/rank-one-power-weakly-mixing-non-singular-transformations-ixbo4a6m2i.pdf

Rank-one power weakly mixing non-singular transformations

If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(\lambda)$ saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point $x_0\in M$ for the geodesic flow at which the associated Perron-Frobenius operator $U_{x_0}: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M)$ has a nontrivial invariant $L^2$ function. The proof is based on an explict Duistermaat-Guillemin-Safarov pre-trace formula and von Neumann's ergodic theorem.

Focal points and sup-norms of eigenfunctions

(Une dichotomie uniforme pour des cocycles a valeurs dans SL(2, R) au-dessus d'une dynamique minimale) On considere des cocycles continus a valeurs dans SL(2, R) au-dessus d'un homeomorphisme minimal d'un ensemble compact de dimension finie. On montre que le cocycle generique soit est uniformement hyperbolique, soit possede une croissance sousexponentielle uniforme.

/pdf/a-uniform-dichotomy-for-generic-rm-sl-2-mathbb-r-cocycles-kdg24meqp9.pdf

A uniform dichotomy for generic ${\rm SL}(2,{\mathbb {R}})$ cocycles over a minimal base

S. Alpern has proved that an invertible antiperiodic measurable measure preserving transformation of a Lebesgue probability space can be rep- resented by k towers of heights n1;::: ;nk, with prescribed measures, provided that the heights have greatest common divisor 1. In this paper we give a sim- ple proof of Alpern's theorem. It is elementary in the sense that it involves no limits and uses Kakutani's easy proof of Rokhlin's Lemma.

/pdf/multiple-rokhlin-tower-theorem-a-simple-proof-1ibdjtu9hv.pdf

Multiple Rokhlin Tower Theorem: A Simple Proof

1. Existence of a finite invariant measure 2. Transformations with no Finite Invariant Measure 3. Infinite Ergodic Transformations 4. Three Basic Examples 5. Properties of Various Sequences 6. Isomorphism Invariants 7. Integer Tilings

Weakly Wandering Sequences in Ergodic Theory

A family of infinite measure preserving odometers is presented which exhibit examples ofp-recurrent but notp+1-recurrent ergodic transformations for everyp>1.

Multiple recurrence and infinite measure preserving odometers

/pdf/ergodic-measure-preserving-transformations-of-finite-type-2gjn0l9mwt.pdf

Ergodic Measure Preserving Transformations of Finite Type

A Structure Theorem for n-to-1 Endomorphisms and Existence of Non-Recurrent Measures

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