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Means in complete manifolds: uniqueness and approximation

Marc Arnaudon, Laurent Miclo (2014)

ESAIM: Probability and Statistics

Let M be a complete Riemannian manifold, M ∈ ℕ and p ≥ 1. We prove that almost everywhere on x = (x1,...,xN) ∈ MN for Lebesgue measure in MN, the measure μ ( x ) = N ∑ k = 1 N x k μ ( x ) = 1 N ∑ k = 1 N δ x k has a uniquep–mean ep(x). As a consequence, if X = (X1,...,XN) is a MN-valued random variable with absolutely continuous law, then almost surely μ(X(ω)) has a unique p–mean. In particular if (Xn)n ≥ 1 is an independent sample of an absolutely continuous law in M, then the process ep,n(ω) = ep(X1(ω),...,Xn(ω)) is...

Measures of maximal entropy for random β -expansions

Karma Dajani, Martijn de Vries (2005)

Journal of the European Mathematical Society

Let β > 1 be a non-integer. We consider β -expansions of the form ∑ i = 1 ∞ d i / β i , where the digits ( d i ) i ≥ 1 are generated by means of a Borel map K β defined on { 0 , 1 } ℕ × [ 0 , ⌊ β ⌋ / ( β − 1 ) ] . We show that K β has a unique mixing measure ν β of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ν β the digits ( d i ) i ≥ 1 form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of β -expansions....

Measure-theoretic unfriendly colorings

Clinton T. Conley (2014)

Fundamenta Mathematicae

We consider the problem of finding a measurable unfriendly partition of the vertex set of a locally finite Borel graph on standard probability space. After isolating a sufficient condition for the existence of such a partition, we show how it settles the dynamical analog of the problem (up to weak equivalence) for graphs induced by free, measure-preserving actions of groups with designated finite generating set. As a corollary, we obtain the existence of translation-invariant random unfriendly colorings...

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....

Méthodes de changement d’échelles en analyse complexe

François Berteloot (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Nous mettons en perspective différentes méthodes de changement d’échelles et illustrons leur pertinence en mettant sur pieds des preuves simples et élémentaires de plusieurs théorèmes biens connus en analyse ou géométrie complexe. Les situations abordées sont variées et la plupart des théorèmes démontrés sont des classiques initialement obtenus entre la fin du xixe  et la seconde moitié du xxe  siècle.

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