Bessa, MarioRocha, JorgeVarandas, Paulo2020-02-052020-02-052018http://hdl.handle.net/10400.6/9020In 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.engUniform hyperbolicityPeriodic pointsFinest dominated splittingOseledets splittingLyapunov exponentsjournal article10.1080/14689367.2018.1433818