We prove measurable Livšic theorems for dynamical systems modelled by Markov towers. Our regularity results apply to solutions of cohomological equations posed on Hénon-like mappings and a wide variety of non-uniformly hyperbolic systems. We consider both Hölder cocycles and cocycles with singularities of prescribed order.

For the circular restricted three-body problem of celestial mechanics with small secondary mass, we prove the existence of uniformly hyperbolic invariant sets of non-planar periodic and chaotic almost collision orbits. Poincaré conjectured existence of periodic ones and gave them the name “second species solutions”. We obtain large subshifts of finite type containing solutions of this type.

In Ergodic optimization, one wants to find ergodic measures maximize or minimize the integral of given continuous functions. This has been succefully studied for uniformly hyperbolic systems generic functions by Bousch and Brémon. this paper, we show that several interesting beyond uniform hyperbolicity, any function a unique maximizing measure with zero entropy. some cases, also know full supp...

We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic ...

We describe examples of hyperbolic surface waves and discuss their connection with initial boundary value and discontinuity problems for hyperbolic systems of PDEs that are weakly but not uniformly stable.

— We consider a partially hyperbolic set K on a Riemannian manifold M whose tangent space splits as TKM = Ecu⊕Es, for which the centre-unstable direction E expands non-uniformly on some local unstable disk. We show that under these assumptions f induces a Gibbs-Markov hyperbolic structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need...

We establish rigorous scaling laws for the average bursting time for bubbling bifurcations of an invariant manifold, assuming the dynamics within the manifold to be uniformly hyperbolic. This type of global bifurcation appears in nearly synchronized systems, and is conjectured to be typical among those breaking the invariance of an asymptotically stable hyperbolic invariant manifold. We conside...