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12
Généricité d'exposants de Lyapunov nonnuls pour des produits déterministes de matrices
 Ann. Inst. H. Poincaré Anal. Non Linéarié
, 2000
"... We propose a geometric sucient criterium \a la Furstenberg" for the existence of nonzero Lyapunov exponents for certain linear cocycles over hyperbolic transformations: nonexistence of probability measures on the bers invariant under the cocycle and under the holonomies of the stable and u ..."
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Cited by 37 (10 self)
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We propose a geometric sucient criterium \a la Furstenberg" for the existence of nonzero Lyapunov exponents for certain linear cocycles over hyperbolic transformations: nonexistence of probability measures on the bers invariant under the cocycle and under the holonomies of the stable and unstable foliations of the transformation. This criterium applies to locally constant and to dominated cocycles over hyperbolic sets endowed with an equilibrium state. As a consequence, we get that nonzero exponents exist for an open dense subset of these cocycles, which is also of full Lebesgue measure in parameter space for generic parametrized families of cocycles. This criterium extends to continuous time cocycles obtained by lifting a hyperbolic ow to a projective ber bundle, tangent to some foliation transverse to the bers. Again, nonzero Lyapunov exponents are implied by nonexistence of transverse measures invariant under the holonomy of the foliation. We apply this last re...
The potential point of view for renormalization
 SENSITIVE DEPENDENCE OF GIBBS MEASURES 27
, 2012
"... Abstract. We study the renormalization for potentials defined by where T : X is the dynamics, H : X → X is onetoone and V : X → R is a potential. We explain how this operator is obtained from the usual renormalization operator for maps and why it has a fixed point. For the MannevillePomeau map, ..."
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Abstract. We study the renormalization for potentials defined by where T : X is the dynamics, H : X → X is onetoone and V : X → R is a potential. We explain how this operator is obtained from the usual renormalization operator for maps and why it has a fixed point. For the MannevillePomeau map, f : [0, 1] , close to the fixed and indifferent point 0 we have, H(x) = x 2 and V*:=log f is a fixed point for R. We are interested in characterizing potentials V such that R n (V ) converges to log f . We recover here the importance of the germ close to the fixed indifferent point. For the shift σ in Σ = {0, 1} N we prove that under mild assumptions there exists a unique kind of H. Consequently, there is a unique kind of fixed potentials for R. These are the" Hofbauerlike" potentials. In the last part, we construct a twoparameters family of potentials defined on Σ related to this renormalization procedure. We show they are less regular than the class R(X) introduced in
From local to global equilibrium states: thermodynamic formalism via inducing scheme
, 2013
"... Abstract. We present a method to construct equilibrium states via induction. This method can be used for some nonuniformly hyperbolic dynamical systems and for nonHölder continuous potentials. It allows to prove the occurrence of phase transition. ..."
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Cited by 1 (1 self)
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Abstract. We present a method to construct equilibrium states via induction. This method can be used for some nonuniformly hyperbolic dynamical systems and for nonHölder continuous potentials. It allows to prove the occurrence of phase transition.
CHAOS: BUTTERFLIES ALSO GENERATE PHASE TRANSITIONS AND PARALLEL UNIVERSES
"... Abstract. We exhibit examples of mixing subshifts of finite type and potentials such that there are phase transitions but the pressure is always strictly convex. More surprisingly, we show that the pressure can be analytic on some interval although there exist several equilibrium states. hal0078000 ..."
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Abstract. We exhibit examples of mixing subshifts of finite type and potentials such that there are phase transitions but the pressure is always strictly convex. More surprisingly, we show that the pressure can be analytic on some interval although there exist several equilibrium states. hal00780005, version 1 22 Jan 2013 1.
LARGE DEVIATIONS FOR RETURN TIMES IN NONRECTANGLE SETS FOR AXIOM A DIFFEOMORPHISMS
"... Abstract. For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder se ..."
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Abstract. For Axiom A diffeomorphisms and equilibrium states, we prove a Large deviations result for the sequence of successive return times into a fixed Borel set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who considered cylinder sets of a Markov partition. 1.
Central Limit Theorem for dimension of Gibbs measures for skew expanding maps
, 2009
"... We consider a class of nonconformal expanding maps on the ddimensional torus. For an equilibrium measure of an Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is ..."
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We consider a class of nonconformal expanding maps on the ddimensional torus. For an equilibrium measure of an Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. An unexpected consequence is that when the measure is not absolutely continuous, then half of the balls of radius ε have a measure smaller than ε δ and half of them have a measure larger than ε δ, where δ is the Hausdorff dimension of the measure. We first show that the problem is equivalent to the study of the fluctuations of some Birkhoff sums. Then we use general results from probability theory as the weak invariance principle and random change of time to get our main theorem. Our method also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of onedimensional non uniformly expanding maps.
Birkhoff averages of Poincaré cycles for Axiom A diffeomorphisms
, 2003
"... We study the time of nth return of orbits to some given (union of) rectangle(s) of a Markov partition of an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure (associated to a Hölderian potential). As a byproduc ..."
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We study the time of nth return of orbits to some given (union of) rectangle(s) of a Markov partition of an Axiom A diffeomorphism. Namely, we prove the existence of a scaled generating function for these returns with respect to any Gibbs measure (associated to a Hölderian potential). As a byproduct, we derive precise large deviation estimates and a central limit theorem for Birkhoff averages of Poincaré cycles. We emphasize that we look at the limiting behavior in term of number of visits (the size of the visited set is kept fixed). Our approach relies on the spectral properties of a oneparameter family of induced transfer operators on unstable leaves crossing the visited set. 1 1
LARGE DEVIATION FOR RETURN TIMES IN OPEN SETS FOR AXIOM A DIFFEOMORPHISMS
, 709
"... Abstract. For axiom A diffeomorphisms and equilibrium state, we prove a Large deviation result for the sequence of successive return times into a fixed open set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who where considering cylinde ..."
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Abstract. For axiom A diffeomorphisms and equilibrium state, we prove a Large deviation result for the sequence of successive return times into a fixed open set, under some assumption on the boundary. Our result relies on and extends the work by Chazottes and Leplaideur who where considering cylinder sets of a Markov partition. 1.
Thermodynamic formalism for Lorenz maps
, 2014
"... For a 2dimensional map representing an expanding geometric Lorenz attractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leave ..."
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For a 2dimensional map representing an expanding geometric Lorenz attractor we prove that the attractor is the closure of a union of as long as possible unstable leaves with ending points. This allows to define the notion of good measures, those giving full measure to the union of these open leaves. Then, for any Hölder continuous potential we prove that there exists at most one relative equilibrium state among the set of good measures. Condition yielding existence are given.