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The tiered Aubry set for autonomous Lagrangian functions

Marie-Claude Arnaud (2008)

Annales de l’institut Fourier

Let L : T M be a Tonelli Lagrangian function (with M compact and connected and dim M 2 ). The tiered Aubry set (resp. Mañé set) 𝒜 T ( L ) (resp. 𝒩 T ( L ) ) is the union of the Aubry sets (resp. Mañé sets) 𝒜 ( L + λ ) (resp. 𝒩 ( L + λ ) ) for λ closed 1-form. Then1.the set 𝒩 T ( L ) is closed, connected and if dim H 1 ( M ) 2 , its intersection with any energy level is connected and chain transitive;2.for L generic in the Mañé sense, the sets 𝒜 T ( L ) ¯ and 𝒩 T ( L ) ¯ have no interior;3.if the interior of 𝒜 T ( L ) ¯ is non empty, it contains a dense subset of periodic points.We then give an example...

Topological conjugacy of cascades generated by gradient flows on the two-dimensional sphere

Andrzej Bielecki (2000)

Annales Polonici Mathematici

This article presents a theorem about the topological conjugacy of a gradient dynamical system with a constant time step and the cascade generated by its Euler method. It is shown that on the two-dimensional sphere S² the gradient dynamical flow is, under some natural assumptions, correctly reproduced by the Euler method for a sufficiently small time step. This means that the time-map of the induced dynamical system is globally topologically conjugate to the discrete dynamical system obtained via...

Topological stability theorem for composite mappings

Isao Nakai (1989)

Annales de l'institut Fourier

We prove that generic convergent diagrams of proper smooth mappings are topologically stable. In proving global properties of diagrams we propose a generalization of the concept of singularity for diagrams, and we establish the geometry of composite mappings.

Topologie du feuilletage fortement stable

Françoise Dal'bo (2000)

Annales de l'institut Fourier

Soient X une variété de Hadamard de courbure - 1 et Γ un groupe d’isométries non élémentaire. Nous montrons qu’il y a équivalence entre la non-arithméticité du spectre des longueurs de Γ X , le mélange topologique du flot géodésique et l’existence d’une feuille dense pour le feuilletage fortement stable.

Topology and dynamics of unstable attractors

M. A. Morón, J. J. Sánchez-Gabites, J. M. R. Sanjurjo (2007)

Fundamenta Mathematicae

This article aims to explore the theory of unstable attractors with topological tools. A short topological analysis of the isolating blocks for unstable attractors with no external explosions leads quickly to sharp results about their shapes and some hints on how Conley's index is related to stability. Then the setting is specialized to the case of flows in ℝⁿ, where unstable attractors are seen to be dynamically complex since they must have external explosions.

Trajectories of polynomial vector fields and ascending chains of polynomial ideals

Dmitri Novikov, Sergei Yakovenko (1999)

Annales de l'institut Fourier

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in n and an algebraic hypersurface. The answer is polynomial in the height (the magnitude of coefficients) of the equation and the size of the curve in the space-time, with the exponent depending only on the degree and the dimension.The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned...

Transitive flows on manifolds.

Víctor Jiménez López, Gabriel Soler López (2004)

Revista Matemática Iberoamericana

In this paper we characterize manifolds (topological or smooth, compact or not, with or without boundary) which admit flows having a dense orbit (such manifolds and flows are called transitive) thus fully answering some questions by Smith and Thomas. Name

Transversely affine foliations of some surface bundles over S 1 of pseudo-Anosov type

Hiromichi Nakayama (1991)

Annales de l'institut Fourier

We consider transversely affine foliations without compact leaves of higher genus surface bundles over the circle of pseudo-Anosov type such that the Euler classes of the tangent bundles of the foliations coincide with that of the bundle foliation. We classify such foliations of those surface bundles whose monodromies satisfy a certain condition.

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