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- A dichotomy in area-preserving reversible mapsPublication . Bessa, Mário; Rodrigues, Alexandre A. P.In this paper we study R-reversible area-preserving maps f : M → M on a two-dimensional Riemannian closed manifold M, i.e. diffeomorphisms f such that R ◦ f = f−1 ◦ R where R: M → M is an isometric involution. We obtain a C1-residual subset where any map inside it is Anosov or else has a dense set of elliptic periodic orbits, thus establishing the stability conjecture in this setting. Along the paper we derive the C1-Closing Lemma for reversible maps and other perturbation toolboxes.
- A note on reversibility and Pell equationsPublication . 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.
- Dynamics of conservative Bykov cycles: tangencies, generalized Cocoon bifurcations and elliptic solutionsPublication . 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.
- Generic area-preserving reversible diffeomorphismsPublication . Bessa, Mário; Carvalho, Maria; Rodrigues, Alexandre A. P.Let M be a surface and R : M → M an area-preserving C∞ diffeomorphism which is an involution and whose set of fixed points is a submanifold with dimension one. We will prove that C1 -generically either an area-preserving R-reversible diffeomorphism, is Anosov, or, for μ-almost every x ∈ M, the Lyapunov exponents at x vanish or else the orbit of x belongs to a compact hyperbolic set with an empty interior. We will also describe a nonempty C1- open subset of area-preserving R-reversible diffeomorphisms where for C1-generically each map is either Anosov or its Lyapunov exponents vanish from almost everywhere.
- The role of the saddle-foci on the structure of a bykov attracting setPublication . Bessa, Mário; Carvalho, Maria; Rodrigues, Alexandre A. P.We consider a one-parameter family ( fλ)λ 0 of symmetric vector fields on the three-dimensional sphere whose flows exhibit a heteroclinic network between two saddle-foci inside a global attracting set. More precisely, when λ = 0, there is an attracting heteroclinic cycle between the two equilibria which is made of two 1- dimensional connections together with a 2-dimensional sphere which is both the stable manifold of one saddle-focus and the unstable manifold of the other. After slightly increasing the parameter while keeping the 1-dimensional connections unaltered, the two-dimensional invariant manifolds of the equilibria become transversal, and thereby create homoclinic and heteroclinic tangles. It is known that these newborn structures are the source of a countable union of topological horseshoes, which prompt the coexistence of infinitely many sinks and saddle-type invariant sets for many values of λ. We show that, for every small enough positive parameter λ, the stable and unstable manifolds of the saddle-foci and those infinitely many horseshoes are contained in the global attracting set of fλ; moreover, the horseshoes belong to the heteroclinic class of the equilibria. In addition, we show that the set of chain-accessible points from either of the saddle-foci is chain-stable and contains the closure of the invariant manifolds of the two equilibria.