Summary: In this paper, we revisit uniformly hyperbolic basic sets and the domination of Oseledets splittings at periodic points. We prove that periodic points with simple Lyapunov spectrum are dense in non-trivial basic pieces of \(C^r\)-residual diffeomorphisms on three-dimensional manifolds \((r\geq 1)\). In the case of the \(C^1\)-topology, we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism \(f\) have simple spectrum \(C^1\)-robustly (in which case \(f\) has a finest dominated splitting into one-dimensional sub-bundles and all Lyapunov exponent functions of \(f\) are continuous in the weak\(^\ast\)-topology) or it can be \(C^1\)-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.