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- Topological aspects of incompressible flowsPublication . Bessa, Mário; Torres, Maria Joana; Varandas, PauloIn 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.
- On the entropy of conservative flowsPublication . Bessa, Mário; Varandas, PauloWe obtain a C1-generic subset of the incompressible flows in a closed three-dimensional manifold where Pesin’s entropy formula holds thus establishing the continuous-time version of Tahzibi (C R Acad Sci Paris I 335:1057–1062, 2002). Moreover, in any compact manifold of dimension larger or equal to three we obtain that the metric entropy function and the integrated upper Lyapunov exponent function are not continuous with respect to the C1 Whitney topology. Finally, we establish the C2- genericity of Pesin’s entropy formula in the context of Hamiltonian four-dimensional flows.
- On the periodic orbits, shadowing and strong transitivity of continuous flowsPublication . Bessa, Mário; 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.
- Trivial and simple spectrum for SL(d, ℝ) cocycles with free base and fiber dynamicsPublication . Bessa, Mário; Varandas, PauloLet AC_D(M,SL(d,R)) denote the pairs (f,A) so that f ∈ A ⊂ Diff (M) is a C^1-Anosov transitive diffeomorphisms and A is an SL(d,R) cocycle dominated with respect to f. We prove that open and densely in AC_D(M,SL(d,R)), in appropriate topologies, the pair (f,A) has simple spectrum with respect to the unique maximal entropy measure μ_f. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in Aut_Leb(M) × Lp(M,SL(d,R)).
- Uniform hyperbolicity revisited: index of periodic points and equidimensional cyclesPublication . Bessa, Mário; Rocha, Jorge; Varandas, PauloIn 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.
- Generic hamiltonian dynamicsPublication . Bessa, Mário; Ferreira, Célia; Rocha, Jorge; Varandas, PauloIn this paper we contribute to the generic theory of Hamiltonians by proving that there is a C2-residual R in the set of C2 Hamiltonians on a closed symplectic manifold M, such that, for any H ∈ R, there is a full measure subset of energies e in H(M) such that the Hamiltonian level (H, e) is topologically mixing; moreover these level sets are homoclinic classes.
- Positivity of the top lyapunov exponent for cocycles on semisimple lie groups over hyperbolic basesPublication . Bessa, Mário; Bochi, Jairo; Cambrainha, Michel; Matheus, Carlos; Varandas, Paulo; Xu, DishengA theorem of Viana says that almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents. In this note we extend this result to cocycles on any noncompact classical semisimple Lie group.