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Polynomial decay of correlations for a class of smooth flows on the two torus

Bassam Fayad (2001)

Bulletin de la Société Mathématique de France

Kočergin introduced in 1975 a class of smooth flows on the two torus that are mixing. When these flows have one fixed point, they can be viewed as special flows over an irrational rotation of the circle, with a ceiling function having a power-like singularity. Under a Diophantine condition on the rotation’s angle, we prove that the special flows actually have a t - η -speed of mixing, for some η > 0 .

Random orderings and unique ergodicity of automorphism groups

Omer Angel, Alexander S. Kechris, Russell Lyons (2014)

Journal of the European Mathematical Society

We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner andWeiss’s example of the group of all permutations...

Random permutations and unique fully supported ergodicity for the Euler adic transformation

Sarah Bailey Frick, Karl Petersen (2008)

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

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.

Ratner's property for special flows over irrational rotations under functions of bounded variation. II

Adam Kanigowski (2014)

Colloquium Mathematicae

We consider special flows over the rotation on the circle by an irrational α under roof functions of bounded variation. The roof functions, in the Lebesgue decomposition, are assumed to have a continuous singular part coming from a quasi-similar Cantor set (including the devil's staircase case). Moreover, a finite number of discontinuities is allowed. Assuming that α has bounded partial quotients, we prove that all such flows are weakly mixing and enjoy the weak Ratner property. Moreover, we provide...

Robust transitivity in hamiltonian dynamics

Meysam Nassiri, Enrique R. Pujals (2012)

Annales scientifiques de l'École Normale Supérieure

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce C r open sets ( r = 1 , 2 , , ) of symplectic diffeomorphisms and Hamiltonian systems, exhibitinglargerobustly transitive sets. We show that the C closure of such open sets contains a variety of systems, including so-calleda priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of...

Semisimple extensions of irrational rotations

Mariusz Lemańczyk, Mieczysław K. Mentzen, Hitoshi Nakada (2003)

Studia Mathematica

We show that semisimple actions of l.c.s.c. Abelian groups and cocycles with values in such groups can be used to build new examples of semisimple automorphisms (ℤ-actions) which are relatively weakly mixing extensions of irrational rotations.

Sequence entropy and rigid σ-algebras

Alvaro Coronel, Alejandro Maass, Song Shao (2009)

Studia Mathematica

We study relationships between sequence entropy and the Kronecker and rigid algebras. Let (Y,,ν,T) be a factor of a measure-theoretical dynamical system (X,,μ,T) and S be a sequence of positive integers with positive upper density. We prove there exists a subsequence A ⊆ S such that h μ A ( T , ξ | ) = H μ ( ξ | ( X | Y ) ) for all finite partitions ξ, where (X|Y) is the Kronecker algebra over . A similar result holds for rigid algebras over . As an application, we characterize compact, rigid and mixing extensions via relative sequence...

Sequences of algebraic integers and density modulo  1

Roman Urban (2007)

Journal de Théorie des Nombres de Bordeaux

We prove density modulo 1 of the sets of the form { μ m λ n ξ + r m : n , m } , where λ , μ is a pair of rationally independent algebraic integers of degree d 2 , satisfying some additional assumptions, ξ 0 , and r m is any sequence of real numbers.

Simple systems are disjoint from Gaussian systems

Andrés del Junco, Mariusz Lemańczyk (1999)

Studia Mathematica

We prove the theorem promised in the title. Gaussians can be distinguished from simple maps by their property of divisibility. Roughly speaking, a system is divisible if it has a rich supply of direct product splittings. Gaussians are divisible and weakly mixing simple maps have no splittings at all so they cannot be isomorphic. The proof that they are disjoint consists of an elaboration of this idea, which involves, among other things, the notion of virtual divisibility, which is, more or less,...

Stability conditions of a queueing system model via fluid limits

Amina Angelika Bouchentouf (2013)

Applicationes Mathematicae

We study the ergodicity of a multi-class queueing model via fluid limits which have the advantage of describing the model in macroscopic form. We consider a model of processing bandwidth requests. Our system is defined by a network of capacity C=N, and a queue which contains an infinite number of items of various sizes 1, a' and b' with 1 < a' < b' < N. The problem considered is: Under what conditions on the parameters of some large classes of networks, do they reach the stationary regime?...

Stable ergodicity and julienne quasi-conformality

Charles Pugh, Michael Shub (2000)

Journal of the European Mathematical Society

In this paper we dramatically expand the domain of known stably ergodic, partially hyperbolic dynamical systems. For example, all partially hyperbolic affine diffeomorphisms of compact homogeneous spaces which have the accessibility property are stably ergodic. Our main tools are the new concepts – julienne density point and julienne quasi-conformality of the stable and unstable holonomy maps. Julienne quasi-conformal holonomy maps preserve all julienne density points.

Statistical properties of unimodal maps

Artur Avila, Carlos Gustavo Moreira (2005)

Publications Mathématiques de l'IHÉS

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition...

Stretching the Oxtoby-Ulam Theorem

Ethan Akin (2000)

Colloquium Mathematicae

On a manifold X of dimension at least two, let μ be a nonatomic measure of full support with μ(∂X) = 0. The Oxtoby-Ulam Theorem says that ergodicity of μ is a residual property in the group of homeomorphisms which preserve μ. Daalderop and Fokkink have recently shown that density of periodic points is residual as well. We provide a proof of their result which replaces the dependence upon the Annulus Theorem by a direct construction which assures topologically robust periodic points.

Strongly mixing sequences of measure preserving transformations

Ehrhard Behrends, Jörg Schmeling (2001)

Czechoslovak Mathematical Journal

We call a sequence ( T n ) of measure preserving transformations strongly mixing if P ( T n - 1 A B ) tends to P ( A ) P ( B ) for arbitrary measurable A , B . We investigate whether one can pass to a suitable subsequence ( T n k ) such that 1 K k = 1 K f ( T n k ) f d P almost surely for all (or “many”) integrable f .

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