Center of Mathematics and Applications of University of Beira Interior

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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

Stretching generic pesin’s entropy formula

Publication . Bessa, Mário; Silva, César M.; Vilarinho, Helder

We prove that Pesin’s entropy formula holds generically within a broad subset of volume-preserving bi-Lipschitz homeomorphisms with respect to the Lipschitz–Whitney topology.

2018Journal article

Open access

The Lyapunov exponents of generic skew-product compact semiflows

Publication . Bessa, Mário; Carvalho, Glória Ferreira

Let F_K denote the set of infinite-dimensional cocycles over a μ-ergodic flow φ^t : M → M and with fiber dynamics given by a compact semiflow on a Hilbert space. We prove that there exists a residual subset R of F_K such that for 􏰤 φ ∈ R and for μ-almost every x ∈ M, either: (i) the limit operator lim (φ^t(x)^*φ^t(x))^(1/2t) when t→∞ is the null operator or else (ii) the Oseledets–Ruelle splitting of 􏰤 along the φ^t -orbit of x has a dominated splitting.

2019Journal article

Restricted access

Sobolev homeomorphisms are dense in volume preserving automorphisms

Publication . Azevedo, Assis; Azevedo, Davide; Bessa, Mário; Torres, Maria Joana

In this paper we prove a weak version of Lusin’s theorem for the space of Sobolev-(1,p) volume preserving homeomor- phisms on closed and connected n-dimensional manifolds, n ≥ 3, for p < n − 1. We also prove that if p > n this result is not true. More precisely, we obtain the density of Sobolev-(1,p) homeomorphisms in the space of volume pre- serving automorphisms, for the weak topology. Furthermore, the regularization of an automorphism in a uniform ball cen- tered at the identity can be done in a Sobolev-(1, p) ball with the same radius centered at the identity.

2019Journal article

Open access

A note on expansiveness and hyperbolicity for generic geodesic flows

Publication . Bessa, Mário

In this short note we contribute to the generic dynamics of geodesic flows associated to metrics on compact Riemannian manifolds of dimension ≥ 2. We prove that there exists a C2-residual subset R of metrics on a given compact Riemannian manifold such that if g∈R, then its associated geodesic flow φ_g(t) is expansive if and only if the closure of the set of periodic orbits of φgt is a uniformly hyperbolic set. For surfaces, we obtain a stronger statement: there exists a C2-residual R such that if g ∈ R, then its associated geodesic flow φgt is expansive if and only if φ_g(t) is an Anosov flow.

2018Journal article

Open access