Uniform hyperbolicity along periodic orbits

Abstract

We introduce the notion of uniform hyperbolicity along periodic orbits (UHPO) for homoclinic classes and provide equivalent conditions under which the UHPO property on a C 1-generic homoclinic class implies hyperbolicity. It is shown that for a C 1-generic locally maximal homoclinic class the UHPO property is equivalent to the non-existence of zero Lyapunov exponents. Using the notion of UHPO, we also give new proofs for some recent C 1-dichotomy theorems.

Connectedness of the set of central Lyapunov exponents

“…for any periodic point q ∈ H(p, f ) with index(q) = index(p), we have q ∼ p, ( [7,8]), (4) for any ergodic measure µ of f , there is a sequence of periodic point p n such that µ pn → µ in weak * topology and O(p n ) → Supp(µ) in Housdorff metric ( [1]). ( 5) If a homoclinic class H(p, f ) is bi-Lyapunov stable then for any sufficiently close g, H(p g , g) is bi-Lyapunov stable, where p g is the continuation of p.…”