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Periodic orbits and chain-transitive sets of C1-diffeomorphisms

Sylvain Crovisier (2006)

Publications Mathématiques de l'IHÉS

We prove that the chain-transitive sets of C1-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C1-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C1-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology...

Periods of Morse-Smale diffeomorphisms of 𝕊²

Juan Luis García Guirao, Jaume Llibre (2008)

Colloquium Mathematicae

The aim of this paper is to describe the set of periods of a Morse-Smale diffeomorphism of the two-dimensional sphere according to its homotopy class. The main tool for proving this is the Lefschetz fixed point theory.

Persistence of fixed points under rigid perturbations of maps

Salvador Addas-Zanata, Pedro A. S. Salomão (2014)

Fundamenta Mathematicae

Let f: S¹ × [0,1] → S¹ × [0,1] be a real-analytic diffeomorphism which is homotopic to the identity map and preserves an area form. Assume that for some lift f̃: ℝ × [0,1] → ℝ × [0,1] we have Fix(f̃) = ℝ × 0 and that f̃ positively translates points in ℝ × 1. Let f ̃ ϵ be the perturbation of f̃ by the rigid horizontal translation (x,y) ↦ (x+ϵ,y). We show that F i x ( f ̃ ϵ ) = ∅ for all ϵ > 0 sufficiently small. The proof follows from Kerékjártó’s construction of Brouwer lines for orientation preserving homeomorphisms...

Pourquoi les points périodiques des homéomorphismes du plan tournent-ils autour de certains points fixes ?

Patrice Le Calvez (2008)

Annales scientifiques de l'École Normale Supérieure

Soit f un homéomorphisme du plan qui préserve l’orientation et qui a un point périodique z * de période q ≥ 2 . Nous montrons qu’il existe un point fixe z tel que le nombre d’enlacement de z * et z ne soit pas nul. En d’autres termes, le nombre de rotation de l’orbite de z * dans l’anneau ℝ 2 ∖ { z } est un élément non nul de ℝ / ℤ . Ceci donne une réponse positive à une question posée par John Franks.

Reducing the number of periodic points in the smooth homotopy class of a self-map of a simply-connected manifold with periodic sequence of Lefschetz numbers

Grzegorz Graff, Agnieszka Kaczkowska (2013)

Annales Polonici Mathematici

Let f be a smooth self-map of an m-dimensional (m ≥ 4) closed connected and simply-connected manifold such that the sequence L ( f ⁿ ) n = 1 ∞ of the Lefschetz numbers of its iterations is periodic. For a fixed natural r we wish to minimize, in the smooth homotopy class, the number of periodic points with periods less than or equal to r. The resulting number is given by a topological invariant J[f] which is defined in combinatorial terms and is constant for all sufficiently large r. We compute J[f] for self-maps...

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