Sectional-Anosov flows

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.     

Sectional-Anosov flows

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.     

On the sensitivity of sectional-Anosov flows

We study sectional-Anosov flows on compact 3-manifolds for which the maximal invariant and nonwandering sets coincide. We prove that every vector field close to one of these flows is sensitive with respect to initial conditions.     

Sectional-Anosov flows on certain compact 3-manifolds

A sectional-Anosov flow is a flow for which the maximal invariant set is sectional-hyperbolic. A generalized 3-handlebody is a compact manifold which is built from a 3-disc attaching 0, 1, 2 and 3-handles at its boundary, one at a time, by attaching maps. We prove that there exist a class of orientable generalized 3-handlebodies supporting sectional-Anosov flows, moreover this class of manifolds is strictly large than the previous one studied in [14].     

One-parameter flows with the pseudo orbit tracing property

A notion of the pseudo orbit tracing property (abbr. POTP) for oneparameter flows on compact metric spaces is discussed. The strong POTP, the strong finite POTP, the normal POTP, the weak POTP and the finite POTP for flows are defined, and the relation between these POTP's is clarified; these are equivalent to each other for flows with no fixed points, but that is not true for flows with fixed points. Moreover, the following is proved; (i) The restriction to the nonwandering set of a flow with the strong POTP has the strong POTP. (ii) If an expansive flow on the nonwandering set has the finite POTP, the flow splits into the finite union of subsystems with topological transitivity. (iii) Every isometric flow with the finite POTP is minimal. (iv) The direct product of flows with the strong POTP does not necessarily have the finite POTP.     

Dynamics of Polynomial Automorphisms of C~N

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Necessary and sufficient conditions for the topological conjugacy of surface diffeomorphisms with a finite number of orbits of heteroclinic tangency

We consider diffeomorphisms of orientable surfaces with the nonwandering set consisting of a finite number of hyperbolic fixed points and the wandering set containing a finite number of heteroclinic orbits of transversal and nontransversal intersection. We distinguish a meaningful class of diffeomorphisms and present a complete topological invariant for this class. The invariant is a scheme consisting of a set of numerical parameters and a set of geometric objects.     

Ergodic properties of Anosov maps with rectangular holes

We study Anosov diffeomorphisms on manifolds in which some holes are cut. The points that are mapped into those holes disappear and never return. The holes studied here are rectangles of a Markov partition. Such maps generalize Smale's horseshoes and certain open billiards. The set of nonwandering points of a map of this kind is a Cantor-like set calledrepeller. We construct invariant and conditionally invariant measures on the sets of nonwandering points. Then we establish ergodic, statistical, and fractal properties of those measures.To the memory of Ricardo MañéPartially supported by NSF grant DMS-9401417.Partially supported by CONICYT and SCIC, Univ. de la Republica (Uruguay).     

Anosov maps with rectangular holes. Nonergodic cases

We study Anosov diffeomorphisms on manifolds in which some holes are cut. The points that are mapped into our holes will disappear and never return. We study the case where the holes are rectangles of a Markov partition. Such maps with holes generalize Smale's horseshoes and certain open billiards. The set of nonwandering points of our map is a Cantor-like set we call arepeller. In our previous paper, we assumed that the map restricted to the remaining rectangles of the Markov partition is topologically mixing. Under this assumption we constructed invariant and conditionally invariant measures on the sets of nonwandering points. Here we relax the mixing assumption and extend our results to nonmixing and nonergodic cases.To the memory of Ricardo MañéPartially supported by NSF grant DMS-9401417.Partially supported by CONICYT and CSIC, Univ. de la Republica (Uruguay).     

非游荡算子的性质*

本文研究一类具有混纯性质的线性算子:非游荡算子,该类算子仅在无穷维线性空间中.我们给出非游荡算子紧集上的超循环分解.     

The recurrence and the gradient-like structure of a flow

In this paper, some new properties of the nonwandering set, the partial order set, the chain recurrent set and the generalized recurrent set of a flow are proved. A new characterization of the gradient-like part of a flow is given. Some results of Conley are generalized.     

Dynamically ordered energy function for Morse-Smale diffeomorphisms on 3-manifolds

This paper deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from the authors?? previous studies where the gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well the dynamics of a diffeomorphism. The paper is devoted to finding conditions for the existence of such an energy function, that is, a function whose set of critical points coincides with the nonwandering set of the considered diffeomorphism. We show that necessary and sufficient conditions for the existence of a dynamically ordered energy function reduce to the type of the embedding of one-dimensional attractors and repellers, each of which is a union of zeroand one-dimensional unstable (stable) manifolds of periodic orbits of a given Morse-Smale diffeomorphism on a closed 3-manifold.     

On the structure of the set of nonwandering points of a pair of coupled quadratic maps

In the plane of parameters, we indicate values for which plane endomorphisms constructed by coupling two identical one-dimensional unimodal quadratic maps have an absorbing domain that contains an attractor and a nontrivial invariant subset of the set of nonwandering points. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 12, pp. 1704–1709, December, 1999.     

Topological aspects of dynamical systems on manifolds

Necessary and sufficient conditions are presented for the existence of dynamical systems on manifolds, for which the set of nonwandering points consists of a disconnected union of 2-dimensional tori with a hyperbolic structure.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 876–878, June, 1993.This paper was supported by the Foundation of Fundamental Researches of the Ukrainian State Committee for Science and Technology.     

On the periodic points of a two-parameter family of maps of the plane

We study the dynamics of a two-parameter family of noninvertible maps of the plane, derived from a model in population dynamics. We prove that, as one parameter varies with the other held fixed, the nonwandering set changes from the empty set to an unstable Cantor set on which the map is topologically equivalent to the shift endomorphism on two symbols. With the help of some numerical work, we trace the genealogies of the periodic points of the family of period 5, and describe their stability types and bifurcations. Among our results we find that the family has a fixed point which undergoes fold, flip and Hopf bifurcations, and that certain families of period five points are interconnected through a codimension-two cusp bifurcation.     

Dynamics of a two parameter family of plane birational maps: Maximal entropy

We mix combinatorial with complex methods to study the dynamics of a real two parameter family of plane birational maps. Specifically, we consider the action of the maps on the Picard group of an appropriate compactification of the complex plane, on the homology groups of a forward invariant real subset of this compactification, and on a Markov partition of the real plane determined by the critical set. For the range of parameters considered, the three actions are equivalent. This allows us to construct a measure of maximal entropy on the real nonwandering set, and it allows us to show that all wandering points are attracted to infinity in a well-defined fashion.     

The topological Markov chain

The topological Markov chain or the subshift of finite type is a restriction of the shift on an invariant subset determined by a 0, 1-matrix, which has some important applications in the theory of dynamical systems. In this paper, the topological Markov chain has been discussed. First, we introduce a structure of the directed gragh on a 0, 1-matrix, and then by using it as a tool, we give some equivalent conditions with respect to the relationship among topological entropy, chaos, the nonwandering set, the set of periodic points and the 0, 1-matrix involved. This work is supported in part by the Foundation of Advanced Research Centre, Zhongshan University.     

Complicated behavior in cubic Hénon maps

Theoretical and Mathematical Physics - We study a generalized Hénon map in two-dimensional space. We find a region of the phase space where the nonwandering set exists, specify parameter...     

Critical sets and unimodal mappings of the square

A lower estimate for the number of different invariant subsets of the set of nonwandering points for a class of unimodal mappings is given. Sufficient conditions for such a mapping to have periodic points of arbitrarily large period are described. The machinery of the appearance of such points may be of very different nature. The existence of mappings with trajectory behavior chaotic in the Li-York sense is established. Conditions for the domain of these trajectories to be arbitrary small are given. Therefore, such trajectories cannot be found by numerical methods. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 669–680, November, 1995.