# Mario BessaUniversidade da Beira Interior | UBI · Department of Mathematics

Mario Bessa

Associate Professor

## About

80

Publications

4,141

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437

Citations

Citations since 2016

Introduction

Additional affiliations

June 2005 - July 2011

## Publications

Publications (80)

We prove the Markus–Yamabe conjecture for compact gradient systems on infinite dimensional Hilbert spaces.

We show that there exists a $C^2$ open dense set of convex bodies with smooth boundary whose billiard map exhibits a non-trivial hyperbolic basic set. As a consequence billiards in generic convex bodies have positive topological entropy and exponential growth of the number of periodic orbits.

We say that a convex planar billiard table \(B\) is \(C^{2}\)-stably expansive on a fixed open subset \(U\) of the phase space if its billiard map \(f_{B}\) is expansive on the maximal invariant set \(\Lambda_{B,U}=\bigcap_{n\in\mathbb{Z}}f^{n}_{B}(U)\), and this property holds under \(C^{2}\)-perturbations of the billiard table.
In this note we pr...

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...

We prove that given a non-wandering point of a Sobolev- ( 1 , p ) (1,p) homeomorphism we can create closed trajectories by making arbitrarily small perturbations. As an application, in the planar case, we obtain that generically the closed trajectories are dense in the non-wandering set.

We prove that the closure of the closed orbits of a generic geodesic
ow on a closed Riemannian n>=2 dimensional manifold is a uniformly hyperbolic set if the shadowing property holds C2-robustly on the metric. We obtain analogous results using weak specification. Finally, if C2-robustly on a generic metric we have the shadowing property allowing b...

We begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

We consider a one-parameter family \((f_\lambda )_{\lambda \, \geqslant \, 0}\) 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 \(\lambda = 0\), there is an attracting heteroclinic cycle between the two equilibria which...

In this short note we obtain a canonical form for commuting divergence-free vector fields.

Let \({\mathscr {F}}_K\) denote the set of infinite-dimensional cocycles over a \(\mu \)-ergodic flow \(\varphi ^t:M\rightarrow M\) and with fiber dynamics given by a compact semiflow on a Hilbert space. We prove that there exists a residual subset \({\mathscr {R}}\) of \({\mathscr {F}}_K\) such that for \(\Phi \in {\mathscr {R}}\) and for \(\mu \)...

In this paper we prove a weak version of Lusin's theorem for the space of Sobolev-(1,p) volume preserving homeomorphisms 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 preserving automorph...

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.

We 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 [7]. 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-...

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 \(\mathscr{R}\) of metrics on a given compact Riemannian manifold such that if \(g\in \mathscr{R}\), then its associated geodesic flow \({\varphi ^{t}_{g}}...

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 ($r\ge 1$). In the case of the $C^1$-topology we can prove that eit...

In this note, we prove the flowbox theorem for divergence-free Lipschitz vector fields.

A 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.

We prove that the geodesic flow on closed surfaces displays a hyperbolic set if the shadowing property holds C2-robustly on the metric. Similar results are obtained when considering even feeble properties like the weak shadowing and the specification properties. Despite the Hamiltonian nature of the geodesic flow, the arguments in the present paper...

We analyze the Lyapunov spectrum of the relative Ruelle operator associated with a skew product whose base is an ergodic automorphism and whose fibers are full shifts. We prove that these operators can be approximated in the C 0-topology by positive matrices with an associated dominated splitting.

In this paper we study R-reversible area-preserving maps f on a
two-dimensional Riemannian closed manifold M, i.e. diffeomorphisms f such that
Ro f=f^{-1}o R where R is an isometric involution on M. We obtain a C1-residual
subset where any map inside it is Anosov or else has a dense set of elliptic
periodic orbits. As a consequence we obtain the pr...

We consider a one-parameter unfolding of a symmetric three-dimensional vector field with a
contracting Bykov network, while preserving the one-dimensional connection. When the parameter is zero,
the dynamics is trivial and well known. After increasing the parameter, the stable and unstable manifolds
of the saddle-foci intersect transversely, creati...

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...

Accepted for publication in Stochastics & Dynamics.

We consider hyperbolic toral automorphisms which are reversible with respect
to a linear area-preserving involution. We will prove that within this context
reversibility is linked to a generalized Pell equation whose solutions we will
analyze. Additionally, we will verify to what extent reversibility is a common
feature and characterize the generic...

Let AC
D
(M,SL(d,ℝ)) denote the pairs (f,A) so that f ∈ A ⊂ Diff1(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 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...

Given a closed Riemannian manifold, we prove the C0-general density theorem
for continuous geodesic flows. More precisely, that there exists a residual (in
the C0-sense) subset of the continuous geodesic flows such that, in that
residual subset, the geodesic flow exhibits dense closed orbits.

In this paper we contribute to the generic theory of Hamiltonians by proving that there is a (Formula presented.)-residual (Formula presented.) in the set of (Formula presented.) Hamiltonians on a closed symplectic manifold (Formula presented.), such that, for any (Formula presented.), there is a full measure subset of energies (Formula presented.)...

For Anosov flows obtained by suspensions of Anosov diffeomorphisms on
surfaces, we show the following type of rigidity result: if a topological
conjugacy between them is differentiable at a point, then the conjugacy has a
smooth extension to the suspended 3-manifold. These result generalize the
similar ones of Sullivan and Ferreira-Pinto for 1-dime...

We will describe the linear two-dimensional Anosov diffeomorphisms which are reversible with respect to a linear area-preserving involution and characterize the generic behavior.

In the present paper we study the C 1-robustness of the three properties: average shadowing, asymptotic average shadowing and limit shadowing within two classes of conservative flows: the incompressible and the Hamiltonian ones. We obtain that the first two properties guarantee dominated splitting (or partial hyperbolicity) on the whole manifold, a...

In the present paper we give a positive answer to a question posed by Viana
on the existence of positive Lyapunov exponents for symplectic cocycles.
Actually, we prove that for an open and dense set of Holder symplectic cocycles
over a non-uniformly hyperbolic diffeomorphism there are non-zero Lyapunov
exponents with respect to any invariant ergodi...

We construct a Hamiltonian suspension for a given symplectomorphism which is the perturbation of a Poincaré map. This is especially useful for the conversion of perturbative results between symplectomorphisms and Hamiltonian flows in any dimension 2d. As an application, using known properties of area-preserving maps, we prove that for any Hamiltoni...

Let M be a closed, symplectic connected Riemannian manifold and f a symplectomorphism on M. We prove that if f is -stably weak shadowable on M, then the whole manifold M admits a partially hyperbolic splitting.

Let
$M$
be a closed
$3$
-dimensional Riemannian manifold. We exhibit a
$C^1$
-residual subset of the set of volume-preserving
$3$
-dimensional flows defined on very general manifolds
$M$
such that, any flow in this residual has zero metric entropy, has zero Lyapunov exponents and, nevertheless, is strongly chaotic in Devaney’s sense. More...

We prove that a C2 Hamiltonian system H in M is globally hyperbolic if any of
the following statements holds: H is robustly topologically stable; H is stably
shadowable; H is stably expansive; and H has the stable weak specification
property. Moreover, we prove that, for a C2-generic Hamiltonian H, the union of
the partially hyperbolic regular ener...

In the present paper we study the C1-robustness of properties: (i) average
shadowing; (ii) asymptotic average shadowing; (iii) limit shadowing. We obtain
that the first two cases guarantee dominated splitting for flows on the whole
manifold, and the third case implies that the flow is a transitive Anosov flow.
We discuss the problems within three c...

We prove that a C1-generic volume-preserving dynamical system (diffeomorphism
or flow) has the shadowing property or is expansive or has the weak
specification property if and only if it is Anosov. Finally, we prove that the
C1-robustness, within the volume-preserving context, of the expansiveness
property and the weak specification property, imply...

We prove that for an open and dense set of Holder symplectic cocycles over a
non-uniformly hyperbolic diffeomorphism there are non-zero Lyapunov exponents
with respect to any invariant ergodic measure with the local product structure.
Moreover, we prove that there exists an open and dense set of Hamiltonian
linear differential systems, over a suspe...

In this paper we generalize [3] and prove that the class of accessible and
saddle-conservative cocycles (a wide class which includes cocycles evolving in
GL(d,R), SL(d,R) and Sp(d,R) Lp-densely have a simple spectrum. We also
generalize [3, 1] and prove that for an Lp-residual subset of accessible
cocycles we have a one-point spectrum, by using a d...

We prove that a Hamiltonian star system, defined on a 2d-dimensional
symplectic manifold M, is Anosov. As a consequence we obtain the proof of the
stability conjecture for Hamiltonians. This generalizes the 4-dimensional
results in [6].

We prove that any C1-stably weakly shadowable volume-preserving
diffeomorphism defined on a compact manifold displays a dominated splitting E +
F. Moreover, both E and F are volume-hyperbolic. Finally, we prove the version
of this result for divergence-free vector fields. As a consequence, in low
dimensions, we obtain global hyperbolicity.

We give a new definition for a Lyapunov exponent (called new Lyapunov
exponent) associated to a continuous map. Our first result states that these
new exponents coincide with the usual Lyapunov exponents if the map is
differentiable. Then, we apply this concept to prove that there exists a
C0-dense subset of the set of the area-preserving homeomorp...

Let M be a closed, symplectic connected Riemannian manifold, f a
symplectomorphism on M. We prove that if f is C1-stably weakly shadowing on M,
then the whole manifold M admits a partially hyperbolic splitting.

In 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 in R, there is an open and dense
set S(H) in H(M) such that, for every e in S(H), the Hamiltonian level (H,e) is
topologically mixing.

In this short note we prove that if a symplectomorphism f is C1-stably
shadowable, then f is Anosov. The same result is obtained for volume-preserving
diffeomorphisms.

We prove that for a C^0-residual set of stochastic matrices over an ergodic
automorphism, the splitting into points with 0 and negative Lyapunov exponent
is dominated. Furthermore, if the Lyapunov spectrum contains at least three
points, then the Oseledets splitting is dominated and, in particular, the
Lyapunov exponents vary continuously. This res...

In this note we show that for any Hamiltonian defined on a symplectic
4-manifold M and any point p in M, there exists a C2-close Hamiltonian whose
regular energy surface through p is either Anosov or it contains a homoclinic
tangency. Our result is based on a general construction of Hamiltonian
suspensions for given symplectomorphisms on Poincar\'e...

Let H be an infinite dimensional separable Hilbert space, X a compact
Hausdorff space and f : X \rightarrow X a homeomorphism which preserves a Borel
ergodic measure which is positive on non-empty open sets. We prove that the
non-uniformly Anosov cocycles are C0-dense in the family of partially
hyperbolic f,H-skew products with non-trivial unstable...

More than thirty years have passed since Newhouse (Am. J. Math. 99:1061–1087, 1977) published a dichotomy on C
1 area-preserving diffeomorphisms. Here we revisit some central results on surface conservative C
1-diffeomorphisms by presenting, in particular, a new proof of Newhouse’s theorem and also by proving some, although folklore,
not yet proved...

We present for a general audience the state of the art on the generic properties of C
2 Hamiltonian dynamical systems.

We prove that the C1 interior of the set of all topologically stable C1
symplectomorphisms is contained in the set of Anosov symplectomorphisms.

We consider the class of C 1 partially hyperbolic vo-lume-preserving flows with one dimensional central direction en-dowed with the C 1 -Whitney topology. We prove that, within this class, any flow can be approximated by an ergodic C 2 volume-preserving flow and so, as a consequence, ergodicity is dense.

We prove that for a C^0-residual of stochastic matrices over an ergodic automorphism the Oseledets splitting is either dominated or the Lyapunov spectrum is trivial. This result extends the dichotomy established by Bochi and Viana to a class of non-accessible cocycles.

We consider a linear differential system of Mathieu equations with periodic
coefficients over periodic closed orbits and we prove that, arbitrarily close
to this system, there is a linear differential system of Hamiltonian damped
Mathieu equations with periodic coefficients over periodic closed orbits such
that, all but a finite number of closed pe...

We prove that the C1-interior of the set of all topologically stable
C1-incompressible flows is contained in the set of Anosov incompressible flows.
Moreover, we obtain an analogous result for the discrete-time case.

The results in Chap. 5 form the basis of a theory of flows on three-dimensional manifolds and paved the way for a global understanding
of the dynamics of C
1 generic flows in dimension 3.
In this chapter we present some results from the generic viewpoint, either for C
1 flows on 3-manifolds, or for C
1 conservative flows on 3-manifolds. This means...

We obtain a C
1-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...

We prove that a volume-preserving three-dimensional flow can be C1-approximated by a volume-preserving Anosov flow or else by another volume-preserving flow exhibiting a homoclinic tangency. This proves the conjecture of Palis for conservative 3-flows and with respect to the C1-topology.

Two of the most popular notions of chaoticity are the one due to Robert Devaney and the one that assumes positive Lyapunov exponents. We discuss the coexistence of both definitions for conservative discrete dynamical systems in the two-sphere and with respect to the C 1 -generic point of view. Editorial remark: There are doubts about a proper peer-...

We show that for any positive forward density subset N \subset Z, there exists an integer m>0, such that, for all n>m, N contains almost perfect n-scaled reproductions of any previously chosen finite set of integers. Comment: 5 pages

Let H be a Hamiltonian, e ∈ H ( M ) ⊂ ℝ and Ɛ H, e a connected component of H ⁻¹ ({ e }) without singularities. A Hamiltonian system, say a triple ( H , e , Ɛ H, e ), is Anosov if Ɛ H, e is uniformly hyperbolic. The Hamiltonian system ( H , e , Ɛ H, e ) is a Hamiltonian star system if all the closed orbits of Ɛ H, e are hyperbolic and the same hold...

A divergence-free vector field satisfies the star property if any divergence-free vector field in some C 1 -neighborhood has all the singularities and all closed orbits hyperbolic. In this article we prove that any divergence-free star vector field defined in a closed three-dimensional manifold is Anosov. Moreover, we prove that a C 1 -structurally...

We prove that a divergence-free and C1-robustly transitive vector field has no singularities. Moreover, if the vector field is smooth enough then the linear Poincaré flow associated to it admits a dominated splitting over M.

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C -residual (dense Gδ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-settin...

We prove the following dichotomy for vector fields in a C1-residual subset of volume-preserving flows: for Lebesgue almost every point all Lyapunov exponents equal to zero or its orbit has a dominated splitting. As a consequence if we have a vector field in this residual that cannot be C1-approximated by a vector field having elliptic periodic orbi...

In this Note we prove that there exists a residual subset of the set of divergence-free vector fields defined on a compact, connected Riemannian manifold M, such that any vector field in this residual satisfies the following property: Given any two nonempty open subsets U and V of M, there exists τ∈R such that Xt(U)∩V≠∅ for any t⩾τ. To cite this ar...

We prove that there exists a residual subset R (with respect to the C 0 topology) of d-dimensional linear differential systems based in a µ-invariant flow and with transition ma-trix evolving in GL(d, R) such that if A ∈ R, then, for µ-a.e. point, the Oseledets splitting along the orbit is dominated (uniform projective hyperbolicity) or else the Ly...

It is well known that an orientation-preserving homeomorphism of the plane without fixed points has trivial dynamics; that is, its non-wandering set is empty and all the orbits diverge to infinity. However, orbits can diverge to infinity in many different ways (or not) giving rise to fundamental regions of divergence. Such a map is topologically eq...

We consider C2 Hamiltonian functions on compact 4-dimensional symplectic manifolds to study elliptic dynamics of the Hamiltonian flow, namely the so-called Newhouse dichotomy. We show that for any open set U intersecting a far from Anosov regular energy surface, there is a nearby Hamiltonian having an elliptic closed orbit through U. Moreover, this...

We prove that there exists an open and dense subset of the incompressible 3-flows of class C^2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincar\'e flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of Bochi-Ma\~n\'e and of Newhou...

We prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point pM that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vecto...

We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdor. space X and preserves a Borel regular ergodic probability which is positive on non-empty open sets, we conclude that there is a C-0-residual subset of cocycles within which,...

We consider a compact 3-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C 1-residual (dense Gδ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flowsetting co...

We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C
2-residual set of Hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being Anosov or having zero Lyapunov exponents almost everywhere. This is in the spirit of the Boch...

Let P be the set of C^1 partially hyperbolic volume-preserving flows with one dimensional central direction endowed with the C^1 flow topology. We prove that any X \in P can be approximated by an ergodic C^2 volume-preserving flow. As a consequence ergodicity is dense in P.

Baraviera and Bonatti proved that it is possible to perturb, in the c^1
topology, a volume-preserving and partial hyperbolic diffeomorphism in order to
obtain a non-zero sum of all the Lyapunov exponents in the central direction.
In this article we obtain the analogous result for volume-preserving flows.

We prove that for a C0-generic (a dense Gδ) subset of all the 2-dimensional conservative nonautonomous linear differential systems, either Lyapunov exponents are zero or there is a dominated splitting μ almost every point.

We consider the set of volume-preserving diffeomor-phisms defined in a d-dimensional (d ≥ 3) closed, connected Rie-mannian manifold and endowed with the C 1 -topology. We prove, generalizing the results in [21, 33], that C 1 -stable expansiveness is tantamount to C 1 -specification and to global hyperbolicity. More-over, we establish that, far from...