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A cut salad of cocycles

Jon Aaronson, Mariusz Lemańczyk, Dalibor Volný (1998)

Fundamenta Mathematicae

We study the centraliser of locally compact group extensions of ergodic probability preserving transformations. New methods establishing ergodicity of group extensions are introduced, and new examples of squashable and non-coalescent group extensions are constructed.

A descriptive view of unitary group representations

Simon Thomas (2015)

Journal of the European Mathematical Society

In this paper, we will study the relative complexity of the unitary duals of countable groups. In particular, we will explain that if G and H are countable amenable non-type I groups, then the unitary duals of G and H are Borel isomorphic.

A map maintaining the orbits of a given ℤ d -action

Bartosz Frej, Agata Kwaśnicka (2016)

Colloquium Mathematicae

Giordano et al. (2010) showed that every minimal free ℤ d -action of a Cantor space X is orbit equivalent to some ℤ-action. Trying to avoid the K-theory used there and modifying Forrest’s (2000) construction of a Bratteli diagram, we show how to define a (one-dimensional) continuous and injective map F on X∖one point such that for a residual subset of X the orbits of F are the same as the orbits of a given minimal free ℤ d -action.

A note on a generalized cohomology equation

K. Krzyżewski (2000)

Colloquium Mathematicae

We give a necessary and sufficient condition for the solvability of a generalized cohomology equation, for an ergodic endomorphism of a probability measure space, in the space of measurable complex functions. This generalizes a result obtained in [7].

A property of ergodic flows

Maria Joiţa, Radu-B. Munteanu (2014)

Studia Mathematica

We introduce a property of ergodic flows, called Property B. We prove that an ergodic hyperfinite equivalence relation of type III₀ whose associated flow has this property is not of product type. A consequence is that a properly ergodic flow with Property B is not approximately transitive. We use Property B to construct a non-AT flow which-up to conjugacy-is built under a function with the dyadic odometer as base automorphism.

Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows

Artur Avila, Marcelo Viana, Amie Wilkinson (2015)

Journal of the European Mathematical Society

We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.

Absolutely continuous dynamics and real coboundary cocycles in L p -spaces, 0 < p < ∞

Ana Alonso, Jialin Hong, Rafael Obaya (2000)

Studia Mathematica

Conditions for the existence of measurable and integrable solutions of the cohomology equation on a measure space are deduced. They follow from the study of the ergodic structure corresponding to some families of bidimensional linear difference equations. Results valid for the non-measure-preserving case are also obtained

Construction of non-constant and ergodic cocycles

Mahesh Nerurkar (2000)

Colloquium Mathematicae

We construct continuous G-valued cocycles that are not cohomologous to any compact constant via a measurable transfer function, provided the underlying dynamical system is rigid and the range group G satisfies a certain general condition. For more general ergodic aperiodic systems, we also show that the set of continuous ergodic cocycles is residual in the class of all continuous cocycles provided the range group G is a compact connected Lie group. The first construction is based on the "closure...

Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich

Raphaël Krikorian (2003/2004)

Séminaire Bourbaki

Étant donnée une fonction régulière de moyenne nulle sur le tore de dimension 2 , il est facile de voir que ses intégrales ergodiques au-dessus d’un flot de translation “générique”sont bornées. Il y a une dizaine d’années, A. Zorich a observé numériquement une croissance en puissance du temps de ces intégrales ergodiques au-dessus de flots d’hamiltoniens (non-exacts) “génériques”sur des surfaces de genre supérieur ou égal à 2 , et Kontsevich et Zorich ont proposé une explication (conjecturelle) de...

Espaces mesurés singuliers fortement ergodiques (Étude métrique–mesurée)

Mikaël Pichot (2007)

Annales de l’institut Fourier

D’après le théorème de Jones-Schmidt, une relation d’équivalence ergodique est fortement ergodique si et seulement si elle ne possède pas de quotient moyennable non trivial. Nous donnons dans cet article deux nouvelles caractérisations de l’ergodicité forte, en termes d’espaces métriques-mesurés. La première identifie ergodicité forte et concentration de la mesure (définie dans ce cadre dans [22]). La seconde caractérise l’existence de quotients moyennables non triviaux par la présence de « suites...

Finitary orbit equivalence and measured Bratteli diagrams

T. Hamachi, M. S. Keane, M. K. Roychowdhury (2008)

Colloquium Mathematicae

We prove a strengthened version of Dye's theorem on orbit equivalence, showing that if the transformation structures are represented as finite coordinate change equivalence relations of ergodic measured Bratteli diagrams, then there is a finitary orbit equivalence between these diagrams.

Isometric extensions, 2-cocycles and ergodicity of skew products

Alexandre Danilenko, Mariusz Lemańczyk (1999)

Studia Mathematica

We establish existence and uniqueness of a canonical form for isometric extensions of an ergodic non-singular transformation T. This is applied to describe the structure of commutors of the isometric extensions. Moreover, for a compact group G, we construct a G-valued T-cocycle α which generates the ergodic skew product extension T α and admits a prescribed subgroup in the centralizer of T α .

Khinchin type condition for translation surfaces and asymptotic laws for the Teichmüller flow

Luca Marchese (2012)

Bulletin de la Société Mathématique de France

We study a diophantine property for translation surfaces, defined in terms of saddle connections and inspired by classical Khinchin condition. We prove that the same dichotomy holds as in Khinchin theorem, then we deduce a sharp estimate on how fast the typical Teichmüller geodesic wanders towards infinity in the moduli space of translation surfaces. Finally we prove some stronger result in genus one.

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