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Statistical properties of unimodal maps

Artur AvilaCarlos 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...

Nonuniform center bunching and the genericity of ergodicity among C 1 partially hyperbolic symplectomorphisms

Artur AvilaJairo BochiAmie Wilkinson — 2009

Annales scientifiques de l'École Normale Supérieure

We introduce the notion of nonuniform center bunching for partially hyperbolic diffeomorphims, and extend previous results by Burns–Wilkinson and Avila–Santamaria–Viana. Combining this new technique with other constructions we prove that C 1 -generic partially hyperbolic symplectomorphisms are ergodic. We also construct new examples of stably ergodic partially hyperbolic diffeomorphisms.

Absolute continuity, Lyapunov exponents and rigidity I: geodesic flows

Artur AvilaMarcelo VianaAmie 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.

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur AvilaMikhail LyubichWeixiao Shen — 2011

Journal of the European Mathematical Society

We study the parameter space of unicritical polynomials f c : z z d + c . For complex parameters, we prove that for Lebesgue almost every c , the map f c is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every c , the map f c is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

Complex one-frequency cocycles

Artur AvilaSvetlana JitomirskayaChristian Sadel — 2014

Journal of the European Mathematical Society

We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other...

Opening gaps in the spectrum of strictly ergodic Schrödinger operators

Artur AvilaJairo BochiDavid Damanik — 2012

Journal of the European Mathematical Society

We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap,...

Exponential mixing for the Teichmüller flow

Artur AvilaSébastien GouëzelJean-Christophe Yoccoz — 2006

Publications Mathématiques de l'IHÉS

We study the dynamics of the Teichmüller flow in the moduli space of abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the S L ( 2 , ) action in the moduli space has a spectral gap.

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