Abstract. We show that for almost every frequency α ∈ R \ Q, for every C ω potential v: R/Z → R, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or non-uniformly hyperbolic (similar results are valid in the smooth category). We describe several applications for the quasiperiodic Schrödinger operator, including persistence of absolutely continuous spectrum under perturbations of the potential. Such results also allow us to complete the proof of the Aubry-André conjecture on the measure of the spectrum of the Almost Mathieu Operator.

Abstract. We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every non-regular parameter is Collet-Eckmann with subexponential recurrence of the critical orbit. Those conditions lead to a detailed and robust statistical description of the dynamics. This proves a version of Palis conjecture in this setting. 1.

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and \complex bounds", two generalized polynomial-like maps which admits a topological conjugacy, quasiconformal outside the lled-in Julia set are, indeed, quasiconformally conjugated. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen

ABSTRACT. We analyse certain parametrized families of one-dimensional maps with infinitely many critical points from the measure-theoretical point of view. We prove that such families have absolutely continuous invariant probability measures for a positive Lebesgue measure subset of parameters. Moreover we show that both the densities of these measures and their entropy vary continuously with the parameter. In addition we obtain sub-exponential rate of mixing for these measures and also that they satisfy the Central Limit Theorem. CONTENTS

Abstract. We obtain estimates relating the phase space and the parameter space of analytic families of unimodal maps. Using those estimates, we show that typical analytic unimodal maps admit a quasiquadratic renormalization. This reduces the study of the statistical properties of typical unimodal maps to the quasiquadratic case which had been studied in [AM2]. The estimates proved here correspond exactly to the Phase-Parameter relation proved in [AM1] in the quadratic case, and allows one to obtain sharp estimates on the dynamics of typical unimodal maps which were available only in the quadratic case: as an example we conclude that the exponent of the polynomial recurrence of the critical orbit is exactly one. We also show that those ideas lead to a new proof of a Theorem of Shishikura: the set of non-renormalizable parameters in the boundary of the Mandelbrot set has Lebesgue measure zero. Further applications of those results can be found in the companion paper

Abstract. The average R(t) = R ϕ dµt of a smooth function ϕ with respect to the SRB measure µt of a smooth one-parameter family ft of piecewise expanding interval maps is not always Lipschitz [4], [17]. We prove that if ft is tangent to the topological class of f, and if ∂tft|t=0 = X ◦ f, then R(t) is differentiable at zero, and R ′(0) coincides with the resummation proposed in [4] of the (a priori divergent) series P∞ R n=0 X(y)∂y(ϕ ◦ fn)(y) dµ0(y) given by Ruelle’s conjecture. In fact, we show that t ↦ → µt is differentiable within Radon measures. It is the first time that a linear response formula is obtained in a setting where structural stability does not hold. Violation of causality [25] reflects the fact that ft may be transversal to the topological class of f. that the set {x ∈ M | limn→ ∞ 1 n 1.

We give a new proof of the hyperbolicity of the fixed point for the period-doubling renormalization operator using the local dynamics near a semi-attractive fixed point (in a Banach space) and the theory of holomorphic motions. We also give a new proof of the exponential contraction of the Feigenbaum renormalization operator in the hybrid class of the period-doubling fixed point: such proof uses the non existence of invariant line fields in the perioddoubling tower (C. McMullen), the topological convergence (D. Sullivan), and a new infinitesimal argument. 1.

Abstract. In the space of C k piecewise expanding unimodal maps, k ≥ 2, we characterize the C 2 smooth families of maps where the topological dynamics does not change (the “smooth deformations”) as the families tangent to a continuous distribution of codimension-one subspaces (the “horizontal” directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of C k−1+Lip deformations tangent to every given C k horizontal direction, for k ≥ 2. 1.

Abstract. We consider quasisymmetric reparametrizations of the parameter space of the quadratic family. We prove that the set of quadratic maps which are either regular or Collet-Eckmann with polynomial recurrence of the critical orbit has full Lebesgue measure. Contents

Abstract. We review recent results that lead to a very precise understanding of the dynamics of typical unimodal maps from the statistical point of view. We also describe the (generalized) renormalization approach to the study of the statistical properties of typical unimodal maps. 1.