Uniform hyperbolicity revisited: index of periodic points and equidimensional cycles

ABSTRACT

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 ≥ 1). In the case of the C¹-topology, we can prove that either all periodic points of a hyperbolic basic piece for a diffeomorphism f  have simple spectrum C¹-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^*-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.

KEYWORDS:

• Uniform hyperbolicity
• periodic points
• finest dominated splitting
• Oseledets splitting
• Lyapunov exponents

2000 MATHEMATICS SUBJECT CLASSIFICATION CODES:

• 37D20
• 37H15
• 37C25
• 37D30

Acknowledgments

MB was partially supported by FCT (‘Fundação para a Ciência e a Tecnologia’), through the Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/MAT/00212/2013. JR was partially supported by by CMUP - (UID/MAT/ 00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020 and by PTDC/MAT-CAL/3884/2014. PV was partially supported by BREUDS and CNPq-Brazil. The authors are grateful to the anonymous referee for the careful reading of the manuscript and many useful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Conselho Nacional de Desenvolvimento Científico e Tecnológico [grant number 301201/2016-1]; FCT (‘Fundação para a Ciência e a Tecnologia’) through the Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior [project number UID/MAT/00212/2013]; CMUP [grant number UID/MAT/ 00144/2013]; FEDER [agreement number PT2020]; PTDC/MAT-CAL/3884/2014; BREUDS; CNPq-Brazil [grant number PQ E-Fomento 301201/2016-1].