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A joint limit theorem for compactly regenerative ergodic transformations

David Kocheim, Roland Zweimüller (2011)

Studia Mathematica

We study conservative ergodic infinite measure preserving transformations satisfying a compact regeneration property introduced by the second-named author in J. Anal. Math. 103 (2007). Assuming regular variation of the wandering rate, we clarify the asymptotic distributional behaviour of the random vector (Zₙ,Sₙ), where Zₙ and Sₙ are respectively the time of the last visit before time n to, and the occupation time of, a suitable set Y of finite measure.

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.

A ratio ergodic theorem for multiparameter non-singular actions

Michael Hochman (2010)

Journal of the European Mathematical Society

We prove a ratio ergodic theorem for non-singular free d and d actions, along balls in an arbitrary norm. Using a Chacon–Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that can concentrate on (thickened) boundaries of balls in d . The proof relies on geometric properties of norms, including the Besicovitch covering lemma and the fact that boundaries of balls have lower dimension than the ambient space. We also show that for general group...

Correlation asymptotics from large deviations in dynamical systems with infinite measure

Sébastien Gouëzel (2011)

Colloquium Mathematicae

We extend a result of Doney [Probab. Theory Related Fields 107 (1997)] on renewal sequences with infinite mean to renewal sequences of operators. As a consequence, we get precise asymptotics for the transfer operator and for correlations in dynamical systems preserving an infinite measure (including intermittent maps with an arbitrarily neutral fixed point).

Ergodic properties of a class of discrete Abelian group extensions of rank-one transformations

Chris Dodd, Phakawa Jeasakul, Anne Jirapattanakul, Daniel M. Kane, Becky Robinson, Noah D. Stein, Cesar E. Silva (2010)

Colloquium Mathematicae

We define a class of discrete Abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that Cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations.

Ergodic transforms associated to general averages

H. Aimar, A. L. Bernardis, F. J. Martín-Reyes (2010)

Studia Mathematica

Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in L p , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in L p . For p = 1 we find that the maximal ergodic...

Ergodicity and conservativity of products of infinite transformations and their inverses

Julien Clancy, Rina Friedberg, Indraneel Kasmalkar, Isaac Loh, Tudor Pădurariu, Cesar E. Silva, Sahana Vasudevan (2016)

Colloquium Mathematicae

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T × T is ergodic, but the product T × T - 1 is not. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

Exact covering maps of the circle without (weak) limit measure

Roland Zweimüller (2002)

Colloquium Mathematicae

We construct maps T on the interval and on the circle which are Lebesgue exact preserving an absolutely continuous infinite measure μ ≪ λ, such that for any probability measure ν ≪ λ the sequence ( n - 1 k = 0 n - 1 ν T - k ) n 1 of arithmetical averages of image measures does not converge weakly.

Infinite ergodic index d -actions in infinite measure

E. Muehlegger, A. Raich, C. Silva, M. Touloumtzis, B. Narasimhan, W. Zhao (1999)

Colloquium Mathematicae

We construct infinite measure preserving and nonsingular rank one d -actions. The first example is ergodic infinite measure preserving but with nonergodic, infinite conservative index, basis transformations; in this case we exhibit sets of increasing finite and infinite measure which are properly exhaustive and weakly wandering. The next examples are staircase rank one infinite measure preserving d -actions; for these we show that the individual basis transformations have conservative ergodic Cartesian...

Infinite measure preserving flows with infinite ergodic index

Alexandre I. Danilenko, Anton V. Solomko (2009)

Colloquium Mathematicae

We construct a rank-one infinite measure preserving flow ( T r ) r such that for each p > 0, the “diagonal” flow ( T r × × T r ) r ( p t i m e s ) on the product space is ergodic.

Invariant measures for piecewise convex transformations of an interval

Christopher Bose, Véronique Maume-Deschamps, Bernard Schmitt, S. Sujin Shin (2002)

Studia Mathematica

We investigate the existence and ergodic properties of absolutely continuous invariant measures for a class of piecewise monotone and convex self-maps of the unit interval. Our assumption entails a type of average convexity which strictly generalizes the case of individual branches being convex, as investigated by Lasota and Yorke (1982). Along with existence, we identify tractable conditions for the invariant measure to be unique and such that the system has exponential decay of correlations on...

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 .

Iterations of the Frobenius-Perron operator for parabolic random maps

Zbigniew S. Kowalski (2009)

Fundamenta Mathematicae

We describe totally dissipative parabolic extensions of the one-sided Bernoulli shift. For the fractional linear case we obtain conservative and totally dissipative families of extensions. Here, the property of conservativity seems to be extremely unstable.

Limit theory for some positive stationary processes with infinite mean

Jon Aaronson, Roland Zweimüller (2014)

Annales de l'I.H.P. Probabilités et statistiques

We prove stable limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from nonintegrable observables over certain piecewise expanding maps. This is done by extending Darling–Kac theory to a suitable family of infinite measure preserving transformations.

Mesures invariantes ergodiques pour des produits gauches

Albert Raugi (2007)

Bulletin de la Société Mathématique de France

Soit ( X , 𝔛 ) un espace mesurable muni d’une transformation bijective bi-mesurable τ . Soit ϕ une application mesurable de X dans un groupe localement compact à base dénombrable G . Nous notons τ ϕ l’extension de τ , induite par ϕ , au produit X × G . Nous donnons une description des mesures positives τ ϕ -invariantes et ergodiques. Nous obtenons aussi une généralisation du théorème de réduction cohomologique de O.Sarig [5] à un groupe LCD quelconque.

Note on the isomorphism problem for weighted unitary operators associated with a nonsingular automorphism

K. Frączek, M. Wysokińska (2008)

Colloquium Mathematicae

We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on ( X ̃ , X ̃ , m ) . Consider the associated unitary operators on L ² ( X ̃ , X ̃ , m ) given by U ̃ S f = ( d ( m S ) / d m ) · ( f S ) and φ · U ̃ S , where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.

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