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Markov partitions for fibre expanding systems

Manfred Denker, Hajo Holzmann (2008)

Colloquium Mathematicae

Fibre expanding systems have been introduced by Denker and Gordin. Here we show the existence of a finite partition for such systems which is fibrewise a Markov partition. Such partitions have direct applications to the Abramov-Rokhlin formula for relative entropy and certain polynomial endomorphisms of ℂ².

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.

Mesures invariantes pour les fractions rationnelles géométriquement finies

Guillaume Havard (1999)

Fundamenta Mathematicae

Let T be a geometrically finite rational map, p(T) its petal number and δ the Hausdorff dimension of its Julia set. We give a construction of the σ-finite and T-invariant measure equivalent to the δ-conformal measure. We prove that this measure is finite if and only if p ( T ) + 1 p ( T ) δ > 2 . Under this assumption and if T is parabolic, we prove that the only equilibrium states are convex combinations of the T-invariant probability and δ-masses at parabolic cycles.

Metastability in the Furstenberg-Zimmer tower

Jeremy Avigad, Henry Towsner (2010)

Fundamenta Mathematicae

According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemerédi’s theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals....

Metric Entropy of Nonautonomous Dynamical Systems

Christoph Kawan (2014)

Nonautonomous Dynamical Systems

We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (Xn, μn) of probability spaces and a sequence of measurable maps fn : Xn → Xn+1 with fnμn = μn+1. This notion generalizes the classical concept of metric entropy established by Kolmogorov and Sinai, and is related via a variational inequality to the topological entropy of nonautonomous systems as defined by Kolyada, Misiurewicz, and Snoha. Moreover, it shares several properties with the classical notion...

Mixing on rank-one transformations

Darren Creutz, Cesar E. Silva (2010)

Studia Mathematica

We prove that mixing on rank-one transformations is equivalent to "the uniform convergence of ergodic averages (as in the mean ergodic theorem) over subsequences of partial sums". In particular, all polynomial staircase transformations are mixing.

Mixing via families for measure preserving transformations

Rui Kuang, Xiangdong Ye (2008)

Colloquium Mathematicae

In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any A , . . . , A k of positive...

Moving averages

S. V. Butler, J. M. Rosenblatt (2008)

Colloquium Mathematicae

In ergodic theory, certain sequences of averages A k f may not converge almost everywhere for all f ∈ L¹(X), but a sufficiently rapidly growing subsequence A m k f of these averages will be well behaved for all f. The order of growth of this subsequence that is sufficient is often hyperexponential, but not necessarily so. For example, if the averages are A k f ( x ) = 1 / ( 2 k ) j = 4 k + 1 4 k + 2 k f ( T j x ) , then the subsequence A k ² f will not be pointwise good even on L , but the subsequence A 2 k f will be pointwise good on L¹. Understanding when the hyperexponential...

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