Centre for Mathematics of the University of Porto

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Publications

Dynamics of conservative Bykov cycles: tangencies, generalized Cocoon bifurcations and elliptic solutions

Publication . Bessa, Mário; Rodrigues, Alexandre A. P.

This paper presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a conservative Bykov cycle where trajectories turn in opposite directions near the two saddle-foci. We show that within the class of divergence-free vector fields that preserve the cycle, tangencies of the invariant manifolds of two hyperbolic saddle-foci densely occur. The global dynamics is persistently dominated by heteroclinic tangencies and by the existence of infinitely many elliptic points coexisting with non-uniformly hyperbolic suspended horseshoes. A generalized version of the Cocoon bifurcations for conservative systems is obtained.

2016Journal article

Open access

Uniform hyperbolicity revisited: index of periodic points and equidimensional cycles

Publication . Bessa, Mário; Rocha, Jorge; Varandas, Paulo

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 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 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.

2018Journal article

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Topological aspects of incompressible flows

Publication . Bessa, Mário; Torres, Maria Joana; Varandas, Paulo

In this article we approach some of the basic questions in topological dynamics, concerning periodic points, transitivity, the shadowing and the gluing orbit properties, in the context of C0 incompressible flows generated by Lipschitz vector fields. We prove that a C0-generic incompressible and fixed-point free flow satisfies the periodic shadowing property, it is transitive and has a dense set of periodic points in the non- wandering set. In particular, a C0-generic fixed-point free incompressible flow satisfies the reparametrized gluing orbit property. We also prove that C0-generic incompressible flows satisfy the general density theorem and the weak shadowing property, moreover these are transitive.

2021Journal article

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A note on reversibility and Pell equations

Publication . Bessa, Mário; Carvalho, Maria; Rodrigues, Alexandre A. P.

This note concerns hyperbolic toral automorphisms which are reversible with respect to a linear area-preserving involution. Due to the low dimension, we will be able to associate the reversibility with a generalized Pell equation from whose set of solutions we will infer further information. Additionally, we will show that reversibility is a rare feature and will characterize the generic setting.

2017Journal article

Open access