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Herman’s last geometric theorem

Bassam Fayad, Raphaël Krikorian (2009)

Annales scientifiques de l'École Normale Supérieure

We present a proof of Herman’s Last Geometric Theorem asserting that if F is a smooth diffeomorphism of the annulus having the intersection property, then any given F -invariant smooth curve on which the rotation number of F is Diophantine is accumulated by a positive measure set of smooth invariant curves on which F is smoothly conjugated to rotation maps. This implies in particular that a Diophantine elliptic fixed point of an area preserving diffeomorphism of the plane is stable. The remarkable...

Heteroclinic solutions for perturbed second order systems

Massimiliano Berti (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The existence of infinitely many heteroclinic orbits implying a chaotic dynamics is proved for a class of perturbed second order Lagrangian systems possessing at least 2 hyperbolic equilibria.

Heterodimensional cycles, partial hyperbolicity and limit dynamics

L. J. Diaz, J. Rocha (2002)

Fundamenta Mathematicae

We study one-parameter families of diffeomorphisms unfolding heterodimensional cycles (i.e. cycles containing periodic points of different indices). We construct an open set of such arcs such that, for a subset of the parameter space with positive relative density at the bifurcation value, the resulting nonwandering set is the disjoint union of two hyperbolic basic sets of different indices and a strong partially hyperbolic set which is robustly transitive. The dynamics of the diffeomorphisms we...

Hierarchy of integrable geodesic flows.

Peter Topalov (2000)

Publicacions Matemàtiques

A family of integrable geodesic flows is obtained. Any such a family corresponds to a pair of geodesically equivalent metrics.

Higher order Schwarzian derivatives in interval dynamics

O. Kozlovski, D. Sands (2009)

Fundamenta Mathematicae

We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive up...

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