The Experts below are selected from a list of 222 Experts worldwide ranked by ideXlab platform
C. A. Morales - One of the best experts on this subject based on the ideXlab platform.
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Transitivity and homoclinic classes for singular-Hyperbolic systems
arXiv: Dynamical Systems, 2020Co-Authors: C. A. Morales, Maria José PacificoAbstract:A singular Hyperbolic Set is a partially Hyperbolic Set with singularities (all Hyperbolic) and volume expanding central direction [MPP1]. We study connected, singularHyperbolic, attracting Sets with dense closed orbits and only one singularity. These Sets are shown to be transitive for most C r flows in the Baire’s second category sence. In general these Sets are shown to be either transitive or the union of two homoclinic classes. In the later case we prove the existence of finitely many homoclinic classes. Our results generalize for singular-Hyperbolic systems a well known result for Hyperbolic systems in [N].
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On the intersection of sectional-Hyperbolic Sets
Journal of Modern Dynamics, 2015Co-Authors: S. Bautista, C. A. MoralesAbstract:We study the intersection of a positively sectional-Hyperbolic Set and a negatively sectional-Hyperbolic Set of a flow on a compact manifold. Indeed, we show that such an intersection is not a Hyperbolic Set in general. Next, we show that such an intersection is a Hyperbolic Set if the Sets involved in the intersection are both transitive. In general, we prove that such an intersection is the disjoint union of a nonsingular Hyperbolic Set, a finite Set of singularities, and a Set of regular orbits joining these dynamical objects. Finally, we exhibit a three-dimensional star flow with a positively (but not negatively) sectional-Hyperbolic homoclinic class and a negatively (but not positively) sectional-Hyperbolic homoclinic class whose intersection is a periodic orbit. This provides a counterexample to a conjecture of Shi, Zhu, Gan and Wen ([25], [26]).
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On the intersection of sectional-Hyperbolic Sets
arXiv: Dynamical Systems, 2014Co-Authors: S. Bautista, C. A. MoralesAbstract:We analyse the intersection of positively and negatively sectional-Hyperbolic Sets for flows on compact manifolds. First we prove that such an intersection is Hyperbolic if the intersecting Sets are both transitive (this is false without such a hypothesis). Next we prove that, in general, such an intersection consists of a nonsingular Hyperbolic Set, finitely many singularities and regular orbits joining them. Afterward we exhibit a three-dimensional star flow with two homoclinic classes, one being positively (but not negatively) sectional-Hyperbolic and the other negatively (but not positively) sectional-Hyperbolic, whose intersection reduces to a single periodic orbit. This provides a counterexample to a conjecture by Shy, Zhu, Gan and Wen (\cite{sgw}, \cite{zgw}).
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On the intersection of homoclinic classes on singular-Hyperbolic Sets
Discrete and Continuous Dynamical Systems, 2007Co-Authors: S. Bautista, C. A. Morales, Maria José PacificoAbstract:We know that two different homoclinic classes contained in the same Hyperbolic Set are disjoint [12]. Moreover, a connected singular-Hyperbolic attracting Set with dense periodic orbits and a unique equilibrium is either transitive or the union of two different homoclinic classes [6]. These results motivate the questions of if two different homoclinic classes contained in the same singular-Hyperbolic Set are disjoint or if the second alternative in [6] cannot occur. Here we give a negative answer for both questions. Indeed we prove that every compact $3$-manifold supports a vector field exhibiting a connected singular-Hyperbolic attracting Set which has dense periodic orbits, a unique singularity, is the union of two homoclinic classes but is not transitive.
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A spectral decomposition for singular-Hyperbolic Sets
Pacific Journal of Mathematics, 2007Co-Authors: C. A. Morales, Maria José PacificoAbstract:We extend the Spectral Decomposition Theorem for Hyperbolic Sets to singular-Hyperbolic Sets on 3-manifolds. We prove that an attracting singular-Hyperbolic Set with dense periodic orbits and a unique equilibrium of a Cr vector field, where r = 1, is a finite union of transitive Sets; the union is disjoint or the Set contains finitely many distinct homoclinic classes. If the vector field is Cr-generic, the union is in fact disjoint.
Todd Fisher - One of the best experts on this subject based on the ideXlab platform.
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Nonlocally maximal and premaximal Hyperbolic Sets
arXiv: Dynamical Systems, 2015Co-Authors: Todd Fisher, T. Petty, S. TikhomirovAbstract:We prove that for any closed manifold of dimension 3 or greater that there is an open Set of smooth flows that have a Hyperbolic Set that is not contained in a locally maximal one. Additionally, we show that the stabilization of the shadowing closure of a Hyperbolic Set is an intrinsic property for premaximality. Lastly,we review some results due to Anosov that concern premaximality.
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The Topology of Hyperbolic Attractors on Compact Surfaces
Ergodic Theory and Dynamical Systems, 2006Co-Authors: Todd FisherAbstract:Suppose M is a compact surface and M is a nontrivial mixing Hyperbolic attractor for some f 2 Di( M). We show that if is a Hyperbolic Set for some g 2 Di( M), then is a nontrivial mixing Hyperbolic attractor or repeller
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Hyperbolic Sets that are not locally maximal
Ergodic Theory and Dynamical Systems, 2006Co-Authors: Todd FisherAbstract:This papers addresses the following topics relating to the structure of Hyperbolic Sets: First, Hyperbolic Sets that are not contained in locally maximal Hyperbolic Sets. Second, the existence of a Markov partition for a Hyperbolic Set. We construct new examples of Hyperbolic Sets which are not contained in locally maximal Hyperbolic Sets. The first example is robust under perturbations and can be constructed on any compact manifold of dimension greater than one. The second example is robust, topologically transitive, and constructed on a 4-dimensional manifold. The third example is volume preserving and constructed on R 4 . We show that every Hyperbolic Set is included in a Hyperbolic Set with a Markov partition. Additionally, we describe a condition that ensures a Hyperbolic Set is included in a locally maximal Hyperbolic Set.
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Hyperbolic Sets with nonempty interior
Discrete and Continuous Dynamical Systems, 2006Co-Authors: Todd FisherAbstract:In this paper we study Hyperbolic Sets with nonempty interior. We prove the folklore theorem that every transitive Hyperbolic Set with interior is Anosov. We also show that on a compact surface every locally maximal Hyperbolic Set with nonempty interior is Anosov. Finally, we give examples of Hyperbolic Sets with nonempty interior for a non-Anosov diffeomorphism.
N. A. Begun - One of the best experts on this subject based on the ideXlab platform.
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On the stability of weakly Hyperbolic invariant Sets
Journal of Differential Equations, 2017Co-Authors: N. A. Begun, V. A. Pliss, George R. SellAbstract:Abstract The dynamical object which we study is a compact invariant Set with a suitable Hyperbolic structure. Stability of weakly Hyperbolic Sets was studied by V. A. Pliss and G. R. Sell (see [1] , [2] ). They assumed that the neutral, unstable and stable linear spaces of the corresponding linearized systems satisfy Lipschitz condition. They showed that if a perturbation is small, then the perturbed system has a weakly Hyperbolic Set K Y , which is homeomorphic to the weakly Hyperbolic Set K of the initial system, close to K, and the dynamics on K Y is close to the dynamics on K. At the same time, it is known that the Lipschitz property is too strong in the sense that the Set of systems without this property is generic. Hence, there was a need to introduce new methods of studying stability of weakly Hyperbolic Sets without Lipschitz condition. These new methods appeared in [16] , [17] , [18] , [19] , [20] . They were based on the local coordinates introduced in [18] and the continuous on the whole weakly Hyperbolic Set coordinates introduced in [19] . In this paper we will show that even without Lipschitz condition there exists a continuous mapping h such that h ( K ) = K Y .
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Perturbations of weakly Hyperbolic invariant Sets of two-dimension periodic systems
Vestnik St. Petersburg University: Mathematics, 2015Co-Authors: N. A. BegunAbstract:The question of structural stability is one of the most important areas in a present-day theory of differential equations. In this paper, we study small C ^1 perturbations of a systems of differential equations. We introduce the concepts of a weakly Hyperbolic invariant Set K and leaf Y for a system of ordinary differential equations. The Lipschitz condition is not assumed. We show that, if the perturbation is small enough, then there is a continuous mapping h , i.e., K → K ^ Y , where K ^ Y is a weakly Hyperbolic Set of the perturbed equation system. Moreover, we show that h ( Y ) is a leaf of the perturbed system.
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On the stability of invariant Sets of leaves of three-dimensional periodic systems
Vestnik St. Petersburg University: Mathematics, 2014Co-Authors: N. A. BegunAbstract:The small C ^1 perturbations of differential equations are studied. The concepts of a weakly Hyperbolic Set K and a leaf ϒ are introduced for a system of ordinary differential equations. The Lipschitz condition is not supposed. It is shown that, if the perturbation is small enough, then there exists a continuous mapping h : ϒ → ϒ^ Y , where ϒ^ Y is a leaf of the perturbed system.
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On the stability of sheet invariant Sets of two-dimensional periodic systems
Vestnik St. Petersburg University: Mathematics, 2012Co-Authors: N. A. BegunAbstract:In the paper small C ^1-perturbations of differential equations are considered. The concepts of a weakly Hyperbolic Set K and a sheet ϒ for a system of ordinary differential equation are introduced. Lipschitz property is not assumed to hold. It is shown that if the perturbation is small enough, then there is a continuous mapping h : ϒ → ϒ^ Y , where ϒ^ Y is a sheet of the perturbed system.
Luis Barreira - One of the best experts on this subject based on the ideXlab platform.
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Dimension of Hyperbolic Sets
Springer Monographs in Mathematics, 2020Co-Authors: Luis BarreiraAbstract:This chapter is dedicated to the study of the dimension of a locally maximal Hyperbolic Set for a conformal flow. We first consider the dimensions along the stable and unstable manifolds and we compute them in terms of the topological pressure. We also show that the Hausdorff dimension and the lower and upper box dimensions of the Hyperbolic Set coincide and that they are obtained by adding the dimensions along the stable and unstable manifolds, plus the dimension along the flow. This is a consequence of the conformality of the flow. The proofs are based on the use of Markov systems.
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Irregular Sets of two-sided Birkhoff averages and Hyperbolic Sets
Arkiv för Matematik, 2016Co-Authors: Luis Barreira, Jinjun Li, Claudia VallsAbstract:For two-sided topological Markov chains, we show that the Set of points for which the two-sided Birkhoff averages of a continuous function diverge is residual. We also show that the Set of points for which the Birkhoff averages have a given Set of accumulation points other than a singleton is residual. A nontrivial consequence of our results is that the Set of points for which the local entropies of an invariant measure on a locally maximal Hyperbolic Set does not exist is residual. This strongly contrasts to the Shannon–McMillan–Breiman theorem in the context of ergodic theory, which says that local entropies exist on a full measure Set.
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The Case of Symbolic Dynamics
Thermodynamic Formalism and Applications to Dimension Theory, 2011Co-Authors: Luis BarreiraAbstract:We consider in this chapter the particular case of symbolic dynamics, which plays an important role in many applications of dynamical systems. In particular, using Markov partitions one can model repellers and Hyperbolic Sets by their associated symbolic dynamics (see Chapters 5 and 6) of dynamical systems, in this case given by a topological Markov chain (also called a subshift of a finite type). Although the codings of a repeller or a Hyperbolic Set need not be invertible (due to the boundaries of the Markov partitions), they still provide sufficient information for the applications in dimension theory and in multifractal analysis of dynamical systems.
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Measures of Maximal Dimension for Hyperbolic Diffeomorphisms
Communications in Mathematical Physics, 2003Co-Authors: Luis Barreira, Christian WolfAbstract:We establish the existence of ergodic measures of maximal Hausdorff dimension for Hyperbolic Sets of surface diffeomorphisms. This is a dimension-theoretical version of the existence of ergodic measures of maximal entropy. The crucial difference is that while the entropy map is upper-semicontinuous, the map ν↦ dim H ν is neither upper-semicontinuous nor lower-semicontinuous. This forces us to develop a new approach, which is based on the thermodynamic formalism. Remarkably, for a generic diffeomorphism with a Hyperbolic Set, there exists an ergodic measure of maximal Hausdorff dimension in a particular two-parameter family of equilibrium measures.
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hausdorff dimension of measures via poincare recurrence
Communications in Mathematical Physics, 2001Co-Authors: Luis Barreira, Benoit SaussolAbstract:We study the quantitative behavior of Poincare recurrence. In particular, for an equilibrium measure on a locally maximal Hyperbolic Set of a C1+α diffeomorphism f, we show that the recurrence rate to each point coincides almost everywhere with the Hausdorff dimension d of the measure, that is, inf{k>0 :fkx∈B(x,r)}∼r−d. This result is a non-trivial generalization of work of Boshernitzan concerning the quantitative behavior of recurrence, and is a dimensional version of work of Ornstein and Weiss for the entropy. We stress that our approach uses different techniques. Furthermore, our results motivate the introduction of a new method to compute the Hausdorff dimension of measures.
D. V. Anosov - One of the best experts on this subject based on the ideXlab platform.
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On Trajectories Entirely Situated Near a Hyperbolic Set
Journal of Mathematical Sciences, 2014Co-Authors: D. V. AnosovAbstract:Let I _1 be a Set of points such that their trajectories under a diffeomorphism f _1 are entirely close enough to a Hyperbolic Set F _1 of this diffeomorphism. Then it is proved that the structure of I _1 and restriction f 1 I 1 $$ {f}_1\left|{}_{I_1}\right. $$ (“motion in I _1”) are essentially defined (up to an equivariant homeomorphism) by “internal dynamics” in F _1 , i.e., by the restriction f 1 F 1 $$ {f}_1\left|{}_{{}_{F_1}}\right. $$ . (In more detail: the equivariant homeomorphism g _1 of the Set F _1 on the Hyperbolic Set F _2 of the second diffeomorphism f _2 (probably, acting on another manifold M _2) is extendable to an equivariant homeomorphic embedding I _1 → M _2 . The image of the imbedding contains all the trajectories f _2 close enough to F _2 . )
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Local maximality of Hyperbolic Sets
Proceedings of the Steklov Institute of Mathematics, 2011Co-Authors: D. V. AnosovAbstract:Two properties of a Hyperbolic Set F are discussed: its local maximality and the property that, in any neighborhood U ⊃ F , there exists a locally maximal Set F ′ that contains F (we suggest calling the latter property local premaximality). Although both these properties of the Set F are related to the behavior of trajectories outside F , it turns out that, in the class of Hyperbolic Sets, the presence or absence of these properties is determined by the interior dynamics on F .
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On certain Hyperbolic Sets
Mathematical Notes, 2010Co-Authors: D. V. AnosovAbstract:Two invariant Sets F of certain diffeomorphisms S that were described by A. Fathi, S. Crovisier, and T. Fisher as examples of Hyperbolic Sets with the property (unexpected at that time) that, in some neighborhood of such an F , there is no locally maximal Set containing F are considered. It is proved that this property, although referring to the behavior of the orbits of S near F , is ultimately determined in the examples mentioned above by a combination of a certain explicitly stated intrinsic property of the action of S on F with the Hyperbolicity of F . (This means that if a Hyperbolic Set F′ for a diffeomorphism S′ is equivariantly homeomorphic to a Fathi-Crovisier or a Fisher Set, then F′ has a neighborhood in which S′ has no locally maximal Set containing F′ .)
