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Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings

Y. Derriennic, K. Frączek, M. Lemańczyk, F. Parreau (2008)

Colloquium Mathematicae

Basic ergodic properties of the ELF class of automorphisms, i.e. of the class of ergodic automorphisms whose weak closure of measures supported on the graphs of iterates of T consists of ergodic self-joinings are investigated. Disjointness of the ELF class with: 2-fold simple automorphisms, interval exchange transformations given by a special type permutations and time-one maps of measurable flows is discussed. All ergodic Poisson suspension automorphisms as well as dynamical systems determined...

Ergodic averages and free 2 actions

Zoltán Buczolich (1999)

Fundamenta Mathematicae

If the ergodic transformations S, T generate a free 2 action on a finite non-atomic measure space (X,S,µ) then for any c 1 , c 2 there exists a measurable function f on X for which ( N + 1 ) - 1 j = 0 N f ( S j x ) c 1 and ( N + 1 ) - 1 j = 0 N f ( T j x ) c 2 µ -almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

Ergodic averages with deterministic weights

Fabien Durand, Dominique Schneider (2002)

Annales de l’institut Fourier

We study the convergence of the ergodic averages 1 N k = 0 N - 1 θ ( k ) f T u k where ( θ ( k ) ) k is a bounded sequence and ( u k ) k a strictly increasing sequence of integers such that Sup α | k = 0 N - 1 θ ( k ) exp ( 2 i π α u k ) | = O ( N δ ) for some δ < 1 . Moreover we give explicit such sequences θ and u and we investigate in particular the case where θ is a q -multiplicative sequence.

Ergodic averages with generalized weights

Doğan Çömez, Semyon N. Litvinov (2006)

Studia Mathematica

Two types of weighted ergodic averages are studied. It is shown that if F = {Fₙ} is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds for F. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are also identified....

Ergodic decomposition of quasi-invariant probability measures

Gernot Greschonig, Klaus Schmidt (2000)

Colloquium Mathematicae

The purpose of this note is to prove various versions of the ergodic decomposition theorem for probability measures on standard Borel spaces which are quasi-invariant under a Borel action of a locally compact second countable group or a discrete nonsingular equivalence relation. In the process we obtain a simultaneous ergodic decomposition of all quasi-invariant probability measures with a prescribed Radon-Nikodym derivative, analogous to classical results about decomposition of invariant probability...

Ergodic properties of group extensions of dynamical systems with discrete spectra

Mieczysław Mentzen (1991)

Studia Mathematica

Ergodic group extensions of a dynamical system with discrete spectrum are considered. The elements of the centralizer of such a system are described. The main result says that each invariant sub-σ-algebra is determined by a compact subgroup in the centralizer of a normal natural factor.

Ergodic properties of skew products with Lasota-Yorke type maps in the base

Zbigniew Kowalski (1993)

Studia Mathematica

We consider skew products T ( x , y ) = ( f ( x ) , T e ( x ) y ) preserving a measure which is absolutely continuous with respect to the product measure. Here f is a 1-sided Markov shift with a finite set of states or a Lasota-Yorke type transformation and T i , i = 1,..., max e, are nonsingular transformations of some probability space. We obtain the description of the set of eigenfunctions of the Frobenius-Perron operator for T and consequently we get the conditions ensuring the ergodicity, weak mixing and exactness of T. We apply these...

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