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A local limit theorem with speed of convergence for euclidean algorithms and diophantine costs

Viviane BaladiAïcha Hachemi — 2008

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

For large , we consider the ordinary continued fraction of =/ with 1≤≤≤, or, equivalently, Euclid’s gcd algorithm for two integers 1≤≤≤, putting the uniform distribution on the set of and s. We study the distribution of the total cost of execution of the algorithm for an additive cost function on the set ℤ of possible digits, asymptotically for →∞. If is nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author. Introducing...

Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane BaladiDaniel 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...

Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Viviane BaladiMasato Tsujii — 2007

Annales de l’institut Fourier

We study spectral properties of transfer operators for diffeomorphisms T : X X on a Riemannian manifold X . Suppose that Ω is an isolated hyperbolic subset for T , with a compact isolating neighborhood V X . We first introduce Banach spaces of distributions supported on V , which are anisotropic versions of the usual space of C p functions C p ( V ) and of the generalized Sobolev spaces W p , t ( V ) , respectively. We then show that the transfer operators associated to  T and a smooth weight g extend boundedly to these spaces, and...

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