6533b7cffe1ef96bd1258ba3

RESEARCH PRODUCT

Dynamic instability in absence of dominated splittings.

subject

description

We want to understand the dynamics in absence of dominated splittings. A dominated splitting is a weak form of hyperbolicity where the tangent bundle splits into invariant subbundles, each of them is more contracted or less expanded by the dynamics than the next one. We first answer an old question from Hirsch, Pugh and Shub, and show the existence of adapted metrics for dominated splittings.Mañé found on surfaces a $C^1$-generic dichotomy between hyperbolicity and Newhouse phenomenons (infinitely many sinks/sources). For that purpose, he showed that without a strong enough dominated splitting along one periodic orbit, a $C^1$-perturbation creates a sink or a source. We generalise that last statement to any dimension, reducing our study to linear cocycles thanks to a Franks' lemma. Abdenur, Bonatti and Crovisier then deduced $C^1$-generic dichotomies in any dimension between Newhouse phenomenons and dominated splittings on the non-wandering sets. The last two chapters are dedicated to creating homoclinic tangencies in absence of stable/unstable dominated splittings, thus extending results of Wen. In the last chapter we finally get that if the homoclinic class of a saddle $P$ has no dominated splitting of same index as $P$, then a perturbation creates a tangency associated to $P$.