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Mario Bessa; Joao Lopes Dias
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We study the dynamical behaviour of Hamiltonian flows defined on 4-dimensional compact symplectic manifolds. We find the existence of a C2-residual set of Hamiltonians for which every regular energy surface is either Anosov or it is in the closure of energy surfaces with zero Lyapunov exponents a.e. This is in the spirit of the Bochi-Mane dichotomy for area-preserving diffeomorphisms on compact surfaces and its continuous-time version for 3-dimensional volume-preserving flows.
Source: http://arxiv.org/abs/0704.3028v2
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Mario Bessa; Jorge Rocha
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We prove that the C1 interior of the set of all topologically stable C1 symplectomorphisms is contained in the set of Anosov symplectomorphisms.
Source: http://arxiv.org/abs/1101.5128v1
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Mario Bessa; Jairo Bochi; Michel Cambrainha; Carlos Matheus; Paulo Varandas; Disheng Xu
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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.
Topics: Dynamical Systems, Mathematics
Source: http://arxiv.org/abs/1611.10158
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Mario Bessa; Jorge Rocha; Paulo Varandas
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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...
Topics: Dynamical Systems, Mathematics
Source: http://arxiv.org/abs/1506.04677
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Mario Bessa; Pedro Duarte
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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 C1-residual such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M.
Source: http://arxiv.org/abs/0709.0700v1
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Vitor Araujo; Mario Bessa
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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 Newhouse for flows with singularities. That is we obtain for a residual subset of the C^1 incompressible flows on 3-manifolds that: (i) either all Lyapunov...
Source: http://arxiv.org/abs/0801.2148v2
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Mario Bessa; Celia Ferreira; Jorge Rocha
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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.
Source: http://arxiv.org/abs/1203.3849v1
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Mário Bessa; Maria Carvalho; Alexandre Rodrigues
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Let M be a surface and R an involution in M whose set of fixed points is a submanifold with dimension 1 and such that R is an isometry. We will show that there is a residual subset of C1 area-preserving R-reversible diffeomorphisms which are either Anosov or have zero Lyapunov exponents at almost every point.
Topics: Dynamical Systems, Mathematics
Source: http://arxiv.org/abs/1402.0070
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Mario Bessa; Maria Carvalho; Alexandre Rodrigues
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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 setting.
Topics: Dynamical Systems, Mathematics
Source: http://arxiv.org/abs/1511.08649
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Mário Bessa; João Lopes Dias
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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 sections already known to yield similar properties.
Source: http://arxiv.org/abs/1107.4286v1
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Mario Bessa; Sandra Vaz
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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.
Source: http://arxiv.org/abs/1203.5139v1
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Mario Bessa; Jorge Rocha
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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.
Source: http://arxiv.org/abs/1006.3725v1
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Mario Bessa; Maria Carvalho
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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.
Source: http://arxiv.org/abs/0909.5667v1
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Mario Bessa; Maria Carvalho
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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 Hausdorff space X and preserves a Borel regular ergodic measure which is positive on non-empty open sets, we conclude that there is a residual subset of cocycles within which, for almost every x, either the Oseledets-Ruelle's decomposition along the orbit of x is dominated or has a trivial spectrum.
Source: http://arxiv.org/abs/0709.3496v1
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Mario Bessa; Manseob Lee; Sandra Vaz
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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.
Source: http://arxiv.org/abs/1207.5546v1
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Mario Bessa; Joao Lopes Dias
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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 implies that for far from Anosov regular energy surfaces of a C2-generic Hamiltonian the elliptic closed orbits are generic.
Source: http://arxiv.org/abs/0801.3072v1
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Mario Bessa; Alexandre Rodrigues
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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 proof of the stability conjecture for this class of maps. Along the paper we also derive the C1-closing lemma for reversible maps and other...
Topics: Mathematics, Dynamical Systems
Source: http://arxiv.org/abs/1403.3572
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Mario Bessa; Celia Ferreira; Jorge Rocha
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A Hamiltonian level, say a pair $(H,e)$ of a Hamiltonian $H$ and an energy $e \in \mathbb{R}$, is said to be Anosov if there exists a connected component $\mathcal{E}_{H,e}$ of $H^{-1}({e})$ which is uniformly hyperbolic for the Hamiltonian flow $X_H^t$. The pair $(H,e)$ is said to be a Hamiltonian star system if there exists a connected component $\mathcal{E}^\star_{H,e}$ of the energy level $H^{-1}({{e}})$ such that all the closed orbits and all the critical points of...
Source: http://arxiv.org/abs/0909.3801v1
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Mario Bessa; Jorge Rocha
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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 orbits, then, there exists a full measure set such that every orbit of this set admits a dominated splitting for the linear Poincare flow. Moreover, we...
Source: http://arxiv.org/abs/0810.3855v1
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Mario Bessa; Helder Vilarinho
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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 different approach of the one given in [3]. Finally, we show that the linear differential system versions of previous results also hold and give some...
Source: http://arxiv.org/abs/1302.6465v1
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Mario Bessa; Paulo Varandas
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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 suspension flow with bounded roof function, displaying at least one positive Lyapunov exponent. In consequence, typical cocycles over a uniformly...
Source: http://arxiv.org/abs/1304.3794v1
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Mario Bessa; Cesar Silva
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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 homeomorphisms defined in a compact, connected and boundaryless surface such that any element inside this residual subset has zero new Lyapunov exponents for...
Source: http://arxiv.org/abs/1004.0641v2
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Mario Bessa; Maria Carvalho
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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 bundles.
Source: http://arxiv.org/abs/1107.3588v1
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Mario Bessa; Paulo Varandas
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Let $AC_D(M,SL(2,\mathbb R))$ denote the pairs $(f,A)$ so that $f\in \mathcal A\subset \text{Diff}^{1}(M)$ is a $C^{1}$-Anosov transitive diffeomorphisms and $A$ is an $SL(2,\mathbb R)$ cocycle dominated with respect to $f$. We prove that open and densely in $AC_D(M,SL(2,\mathbb R))$ (in appropriate topologies) the pair $(f,A)$ has simple spectrum with respect to the unique maximal entropy measure $\mu_f$. On the other hand, there exists a residual subset $\mathcal{R}\subset...
Source: http://arxiv.org/abs/1209.2293v3
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Mario Bessa; Maria Joana Torres
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Given a closed Riemannian manifold, we prove the C1-general density theorem for geodesic flows. More precisely, that C1-generic metrics have dense closed geodesics.
Source: http://arxiv.org/abs/1304.1069v1
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Mario Bessa; Alexandre Rodrigues
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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...
Topics: Dynamical Systems, Mathematics
Source: http://arxiv.org/abs/1511.08648
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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 periodic coefficients, have unstable solutions. The perturbations will be peformed in the periodic coefficients.
Source: http://arxiv.org/abs/1006.5025v1
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Mário Bessa; Manuel Stadlbauer
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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 result extends the dichotomy established by Bochi and Viana to a class of non-accessible cocycles.
Source: http://arxiv.org/abs/1108.3746v2
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Mario Bessa; Paulo Varandas
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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 \cite{T}. 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 $C^1$ Whitney topology. Finally, we establish the $C^2$-genericity of Pesin's entropy formula in...
Source: http://arxiv.org/abs/1002.2366v1
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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.
Source: http://arxiv.org/abs/1112.3466v1
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Mario Bessa; Jorge Rocha
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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.
Source: http://arxiv.org/abs/math/0610558v1
