An averaging method for getting uniformly valid asymptotic approximations of the solution of hyperbolic systems of equations is presented. The averaged system of equations disintegrates into independent equations for non-resonance systems. We consider the resonance conditions for some classes of solutions. The averaged system can be solved numerically in the resonance case. The shallow water pr...

(1) We felt that there was a geometric structure (namely the structure of critical regions) that, if added, could make more transparent the overall purpose of the different components of the very complicated tour de force analysis of Benedicks and Carleson. It could also induce non-trivial technical simplifications. We also felt that the various technical details skipped in [3], considering the...

We obtain large deviation bounds for non-uniformly expanding maps with non-flat singularities or criticalities and for partially hyperbolic non-uniformly expanding attracting sets. That is, given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map, showing that the Lebesgue measure of the set of p...

For Hölder continuous cocycles over an invertible, Lipschitz base, we establish the Hölder continuity of Oseledets subspaces on compact sets of arbitrarily large measure. This extends a result of Araújo et al [On Hölder-continuity of Oseledets subspaces J. Lond. Math. Soc. 93 (2016) 194–218] by considering possibly non-invertible cocycles, which, in addition, may take values in the space of com...

We show that for almost every frequency α ∈ R\Q, for every Cω potential v : R/Z→ R, and for almost every energy E the corresponding quasiperiodic Schrödinger cocycle is either reducible or non-uniformly hyperbolic. This result gives a very good control on the absolutely continuous part of the spectrum of the corresponding quasiperiodic Schrödinger operator, and allows us to complete the proof o...

We consider continuous SL(2, R)-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2, R)-cocycle. Using this, we show that if a cocycle’s homotopy class does not display a certain obstruction to uniform hy...

Pesin Theory-An important branch of dynamical systems and of smooth ergodic theory, with many applications to non-linear dynamics. The name is due to the landmark work of Yakov B. Pesin in the mid-seventies 20, 21, 22]. Sometimes it is also referred to as the theory of smooth dynamical systems with non-uniformly hyperbolic behavior, or simply theory of non-uniformly hyperbolic dynamical systems...

Let Λ be a compact locally maximal invariant set of a Cdiffeomorphism f : M → M on a smooth Riemannian manifold M . In this paper we study the topological pressure Ptop(φ) (with respect to the dynamical system f |Λ) for a wide class of Hölder continuous potentials and analyze its relation to dynamical, as well as geometrical, properties of the system. We show that under a mild nonuniform hyperb...

Pesin Theory-An important branch of dynamical systems and of smooth ergodic theory, with many applications to non-linear dynamics. The name is due to the landmark work of Yakov B. Pesin in the mid-seventies 20, 21, 22]. Sometimes it is also referred to as the theory of smooth dynamical systems with non-uniformly hyperbolic behavior, or simply theory of non-uniformly hyperbolic dynamical systems...

We give sufficient conditions for the uniform hyperbolicity of certain nonuniformly hyperbolic dynamical systems. In particular, we show that local diffeomorphisms that are nonuniformly expanding on sets of total probability are necessarily uniformly expanding. We also present a version of this result for diffeomorphisms with nonuniformly hyperbolic sets.