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- Uniform hyperbolicity revisited: index of periodic points and equidimensional cyclesPublication . Bessa, Mario; Rocha, Jorge; Varandas, PauloIn this paper, we revisit uniformly hyperbolic basic sets and the dom- ination of Oseledets splittings at periodic points. We prove that peri- odic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of Cr-residual diffeomorphisms on three-dimensional manifolds (r & 1). In the case of the C1-topology, we can prove that either all periodic points of a hyperbolic basic piece for a diffeomor- phism f have simple spectrum C1 -robustly (in which case f has a finest dominated splitting into one-dimensional sub-bundles and all Lya- punov exponent functions of f are continuous in the weak∗ -topology) or it can be C1-approximated by an equidimensional cycle associated to periodic points with robust different signatures. The latter can be used as a mechanism to guarantee the coexistence of infinitely many periodic points with different signatures.
- On the periodic orbits, shadowing and strong transitivity of continuous flowsPublication . Bessa, Mario; Torres, Maria Joana; Varandas, PauloWe prove that chaotic flows (i.e. flows that satisfy the shadowing property and have a dense subset of periodic orbits) satisfy a reparametrized gluing orbit property similar to the one introduced in Bomfim and Varandas (2015). In particular, these are strongly transitive in balls of uniform radius. We also prove that the shadowing property for a flow and a generic time-t map, and having a dense subset of periodic orbits hold for a C0-Baire generic subset of Lipschitz vector fields, that generate continuous flows. Similar results also hold for C0-generic homeomorphisms and, in particular, we deduce that chain recurrent classes of C0-generic homeomorphisms have the gluing orbit property.