Asymptotic linking of periodic orbits for diffeomorphisms of the two-disk (original) (raw)
1989
The periodic orbit structure of orientation preserving diffeomorphisms on D2 with topological entropy zero Annales de l'I. H. P., section A, tome 50, n o 3 (1989), p. 335-356 http://www.numdam.org/item?id=AIHPA\_1989\_\_50\_3\_335\_0 © Gauthier-Villars, 1989, tous droits réservés. L'accès aux archives de la revue « Annales de l'I. H. P., section A » implique l'accord avec les conditions générales d'utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
Cascades of periodic orbits in two dimensions
For orientation preserving-homeomorphisms of the 2-disk, we consider the forcing relation among braid types of periodic orbits. We show that if a braid type forces a braid type , then either is an extension of or forces infinitely many extensions of . This generalizes a well known result in one-dimensional dynamics
Annals of Mathematics, 2007
We continue the previous article's discussion of bounds, for prevalent diffeomorphisms of smooth compact manifolds, on the growth of the number of periodic points and the decay of their hyperbolicity as a function of their period n. In that article we reduced the main results to a problem, for certain families of diffeomorphisms, of bounding the measure of parameter values for which the diffeomorphism has (for a given period n) an almost periodic point that is almost nonhyperbolic. We also formulated our results for 1-dimensional endomorphisms on a compact interval. In this article we describe some of the main techniques involved and outline the rest of the proof. To simplify notation, we concentrate primarily on the 1-dimensional case.
Diffeotopically trivial periodic diffeomorphisms
Inventiones Mathematicae, 1970
In this paper, we settle negatively an old question as to whether all free periodic diffeomorphisms that are diffeotopic to the identity can be found by restricting circle group actions to finite cyclic subgroups. More precisely, we construct examples of periodic diffeomorphisms of D 2" for n>3 which on S z"-~ are free and cannot be obtained from any piecewise linear (PL) circle group action on S 2"-1 (cf. Gluck [5]).
Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps
Nonlinearity, 2013
We study dynamics and bifurcations of two-dimensional reversible maps having non-transversal heteroclinic cycles containing symmetric saddle periodic points. We consider one-parameter families of reversible maps unfolding generally the initial heteroclinic tangency and prove that there are infinitely sequences (cascades) of bifurcations of birth of asymptotically stable and unstable as well as elliptic periodic orbits.
Action and periodic orbits on annulus
2021
We consider the classical problem of area-preserving maps on annulus A = S × [0, 1] . Let Mf be the set of all invariant probability measures of an area-preserving, orientation preserving diffeomorphism f on A. Given any μ1 and μ2 in Mf , Franks [2][3], generalizing Poincaré’s last geometric theorem (Birkhoff [1]), showed that if their rotation numbers are different, then f has infinitely many periodic orbits. In this paper, we show that if μ1 and μ2 have different actions, even if they have the same rotation number, then f has infinitely many periodic orbits. In particular, if the action difference is larger than one, then f has at least two fixed points. The same result is also true for area-preserving diffeomorphisms on unit disk, where no rotation number is available.
On the Regularity of the Displacement Sequence of an Orientation Preserving Circle Homeomorphism
2015
We investigate the regularity properties of the displacement sequence () () () (), 2 exp , 1 mod 1 ix z x x z n n n π = Φ − Φ = η − where R R → Φ : is a lift of an orientation preserving circle homeomorphism. If the rotation number () q p = ϕ is rational, then () z n η is asymptotically periodic with semi-period q. This WACŁAW MARZANTOWICZ and JUSTYNA SIGNERSKA 12 convergence to a periodic sequence is uniform in z if we admit that some points are iterated backward instead of taking only forward iterations for all z. This leads to the notion of an basins'-ε edge, which we illustrate by the numerical example. If () , Q ∈ / ϕ then some classical results in topological dynamics yield that the displacement sequence also exhibits some regularity properties, which we define and prove in the second part of the paper.