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A version of non-Hamiltonian Liouville equation

Celina Rom (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we give a version of the theorem on local integral invariants of systems of ordinary differential equations. We give, as an immediate conclusion of this theorem, a condition which guarantees existence of an invariant measure of local dynamical systems. Results of this type lead to the Liouville equation and have been frequently proved under various assumptions. Our method of the proof is simpler and more direct.

Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows

Artur Avila, Marcelo Viana, Amie Wilkinson (2015)

Journal of the European Mathematical Society

We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.

Déviations de moyennes ergodiques, flots de Teichmüller et cocycle de Kontsevich-Zorich

Raphaël Krikorian (2003/2004)

Séminaire Bourbaki

Étant donnée une fonction régulière de moyenne nulle sur le tore de dimension 2 , il est facile de voir que ses intégrales ergodiques au-dessus d’un flot de translation “générique”sont bornées. Il y a une dizaine d’années, A. Zorich a observé numériquement une croissance en puissance du temps de ces intégrales ergodiques au-dessus de flots d’hamiltoniens (non-exacts) “génériques”sur des surfaces de genre supérieur ou égal à 2 , et Kontsevich et Zorich ont proposé une explication (conjecturelle) de...

Equilibrium measures for holomorphic endomorphisms of complex projective spaces

Mariusz Urbański, Anna Zdunik (2013)

Fundamenta Mathematicae

Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space k , k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number κ f > 0 such that if ϕ: J → ℝ is a Hölder continuous function with s u p ( ϕ ) - i n f ( ϕ ) < κ f , then ϕ admits a unique equilibrium state μ ϕ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system ( f , μ ϕ ) is K-mixing, whence ergodic. Proving...

Gibbs states for non-irreducible countable Markov shifts

Andrei E. Ghenciu, Mario Roy (2013)

Fundamenta Mathematicae

We study Markov shifts over countable (finite or countably infinite) alphabets, i.e. shifts generated by incidence matrices. In particular, we derive necessary and sufficient conditions for the existence of a Gibbs state for a certain class of infinite Markov shifts. We further establish a characterization of the existence, uniqueness and ergodicity of invariant Gibbs states for this class of shifts. Our results generalize the well-known results for finitely irreducible Markov shifts.

Gibbs-Markov-Young structures*, **, ***

Carla L. Dias (2012)

ESAIM: Proceedings

We discuss the geometric structures defined by Young in [9, 10], which are used to prove the existence of an ergodic absolutely continuous invariant probability measure and to study the decay of correlations in expanding or hyperbolic systems on large parts.

La mesure d’équilibre d’un endomorphisme de k ( )

Xavier Buff (2004/2005)

Séminaire Bourbaki

Soit f un endomorphisme holomorphe de k ( ) . Je présenterai une construction géométrique, due à Briend et Duval, d’une mesure de probabilité μ ayant les propriétés suivantes : μ reflète la distribution des préimages des points en dehors d’un ensemble exceptionnel algébrique, les points périodiques répulsifs de f s’équidistribuent par rapport à μ et μ est l’unique mesure d’entropie maximale de f .

Linear growth of the derivative for measure-preserving diffeomorphisms

Krzysztof Frączek (2000)

Colloquium Mathematicae

We consider measure-preserving diffeomorphisms of the torus with zero entropy. We prove that every ergodic C 1 -diffeomorphism with linear growth of the derivative is algebraically conjugate to a skew product of an irrational rotation on the circle and a circle C 1 -cocycle. We also show that for no positive β ≠ 1 does there exist an ergodic C 2 -diffeomorphism whose derivative has polynomial growth with degree β.

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi, Daniel Smania (2012)

Annales scientifiques de l'École Normale Supérieure

We consider C 2 families t f t of  C 4 unimodal maps f t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure μ t of  f t depends differentiably on  t , as a distribution of order 1 . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of  μ t for a Benedicks-Carleson map f t , in terms of a single smooth function and the inverse branches...

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