Minimal Morse flows on compact manifolds

2006

Abstract In this paper we prove, using the Poincare–Hopf inequalities, that a minimal number of non-degenerate singularities can be computed in terms only of abstract homological boundary information. Furthermore, this minimal number can be realized on some manifold with non-empty boundary satisfying the abstract homological boundary information. In fact, we present all possible indices and types (connecting or disconnecting) of singularities realizing this minimal number. The Euler characteristics of all manifolds realizing this minimal number are obtained and the associated Lyapunov graphs of Morse type are described and shown to have the lowest topological complexity.

Conjugate unstable manifolds and their underlying geometrized Markov partitions

2000

Abstract Conjugate unstable manifolds of saturated hyperbolic sets of Smale diffeomorphisms are characterized in terms of the combinatorics of their geometrized Markov partitions. As a consequence, the relationship between the local and the global point of view is also made explicit.

Homeomorphisms of the Sierpinski curve with periodic properties

2013

In this paper, we study the three following types of homeomorphisms of the Sierpinski curve of the two sphere : pointwise periodic, periodic, and almost periodic, and we prove that they are equivalent. We show that a subgroup of homeomorphisms whose orbits are all finite, is a finite subgroup.

Integration of a Dirac comb and the Bernoulli polynomials

2016

Abstract For any positive integer n , we consider the ordinary differential equations of the form y ( n ) = 1 − Ш + F where Ш denotes the Dirac comb distribution and F is a piecewise- C ∞ periodic function with null average integral. We prove the existence and uniqueness of periodic solutions of maximal regularity. Above all, these solutions are given by means of finite explicit formulae involving a minimal number of Bernoulli polynomials. We generalize this approach to a larger class of differential equations for which the computation of periodic solutions is also sharp, finite and effective.

Hasse diagrams and orbit class spaces

2011

Abstract Let X be a topological space and G be a group of homeomorphisms of X. Let G ˜ be an equivalence relation on X defined by x G ˜ y if the closure of the G-orbit of x is equal to the closure of the G-orbit of y. The quotient space X / G ˜ is called the orbit class space and is endowed with the natural order inherited from the inclusion order of the closure of the classes, so that, if such a space is finite, one can associate with it a Hasse diagram. We show that the converse is also true: any finite Hasse diagram can be realized as the Hasse diagram of an orbit class space built from a dynamical system ( X , G ) where X is a compact space and G is a finitely generated group of homeomo…

Topological and dynamical classification of the unstable manifolds of one-rectangle systems

2001

On the variations of the Betti numbers of regular levels of Morse flows

2011

Abstract We generalize results in Cruz and de Rezende (1999) [7] by completely describing how the Betti numbers of the boundary of an orientable manifold vary after attaching a handle, when the homology coefficients are in Z, Q, R or Z p Z with p prime. First we apply this result to the Conley index theory of Lyapunov graphs. Next we consider the Ogasa invariant associated with handle decompositions of manifolds. We make use of the above results in order to obtain upper bounds for the Ogasa invariant of product manifolds.

Variétés instables d'ensembles hyperboliques

1999

Resume Les varietes instables de systemes hyperboliques admettant une partition de Markov a un rectangle sont ici caracterisees a homeomorphisme pres et a conjugaison topologique (des dynamiques sous-jacentes) pres. De telles classes seront decrites a l'aide d'objets algebriques naturellement associes aux systemes sous-jacents.

Lyapunov graphs for circle valued functions

2018

International audience; Conley index theory is used to obtain results for flows associated to circular Lyapunov functions defined on general compact smooth n-manifolds. This is done in terms of their underlying circular Lyapunov digraphs, which are generalizations of Morse digraphs, by extensively studying their combinatorics, invariants and realizability.