Diffeomorphisms with an infinite set of stable periodic points

2018 ◽  
Author(s):  
Ekaterina Viktorovna Vasil'eva
Keyword(s):  
Periodic Points ◽  
Stable Periodic ◽  
Infinite Set

scholarly journals Multi-pass stable periodic points of diffeomorphism of a plane with a homoclinic orbit

2021 ◽  
Vol 8 (3) ◽  
pp. 406-416
Author(s):  
Ekaterina V. Vasil’eva ◽  
Keyword(s):  
Finite Order ◽  
Unstable Manifold ◽  
Small Neighborhood ◽  
Periodic Points ◽  
Homoclinic Point ◽  
Hyperbolic Point ◽  
Stable Periodic ◽  
Nontransversal Homoclinic Point ◽  
Stable And Unstable Manifold ◽  
Infinite Set

A diffeomorphism of a plane into itself with a fixed hyperbolic point and a nontransversal point homoclinic to it is studied. There are various ways of touching a stable and unstable manifold at a homoclinic point. Periodic points whose trajectories do not leave the vicinity of the trajectory of a homoclinic point are divided into a countable set of types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns that lie outside a sufficiently small neighborhood of the hyperbolic point. Earlier in the articles of Sh. Newhouse, L. P. Shil’nikov, B. F. Ivanov and other authors, diffeomorphisms of the plane with a nontransversal homoclinic point were studied, it was assumed that this point is a tangency point of finite order. In these papers, it was shown that in a neighborhood of a homoclinic point there can be infinite sets of stable two-pass and three-pass periodic points. The presence of such sets depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point with a finite order of tangency of a stable and unstable manifold. It is shown in the paper that for any fixed natural number n, a neighborhood of a nontransversal homolinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents separated from zero.

scholarly journals Different types of stable periodic points of diffeomorphism of a plane with a homoclinic orbit

2021 ◽  
Vol 8 (2) ◽  
pp. 295-304
Author(s):  
Ekaterina V. Vasil’eva ◽  
Keyword(s):  
Finite Order ◽  
Unstable Manifold ◽  
Periodic Points ◽  
Homoclinic Point ◽  
Double Pass ◽  
Stable And Unstable Manifolds ◽  
Hyperbolic Point ◽  
Stable Periodic ◽  
Stable And Unstable Manifold ◽  
Infinite Set

A diffeomorphism of the plane into itself with a fixed hyperbolic point is considered; the presence of a nontransverse homoclinic point is assumed. Stable and unstable manifolds touch each other at a homoclinic point; there are various ways of touching a stable and unstable manifold. In the works of Sh. Newhouse, L. P. Shilnikov and other authors, studied diffeomorphisms of the plane with a nontranverse homoclinic point, under the assumption that this point is a tangency point of finite order. It follows from the works of these authors that an infinite set of stable periodic points can lie in a neighborhood of a homoclinic point; the presence of such a set depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point at which the tangency of a stable and unstable manifold is a tangency of finite order. Allocate a countable number of types of periodic points lying in the vicinity of a homoclinic point; points belonging to the same type are called n-pass (multi-pass), where n is a natural number. In the present paper, it is shown that if the tangency is not a tangency of finite order, the neighborhood of a nontransverse homolinic point can contain an infinite set of stable single-pass, double-pass, or three-pass periodic points with characteristic exponents separated from zero.

Stable periodic points of infinitely smooth diffeomorphisms

2013 ◽  
Vol 87 (1) ◽  
pp. 3-4
Author(s):  
E. V. Vasil’eva
Keyword(s):  
Periodic Points ◽  
Stable Periodic

Diffeomorphisms of the plane with stable periodic points

2012 ◽  
Vol 48 (3) ◽  
pp. 309-317
Author(s):  
E. V. Vasil’eva
Keyword(s):  
Periodic Points ◽  
Stable Periodic

Homeomorphic restrictions of smooth endomorphisms of an interval

1992 ◽  
Vol 12 (3) ◽  
pp. 429-439 ◽  
Author(s):  
Karen M. Brucks ◽  
Maria Victoria Otero-Espinar ◽  
Charles Tresser
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Periodic Points ◽  
Limit Set ◽  
Asymptotic Dynamics ◽  
Infinite Set

AbstractWe describe the asymptotic dynamics of homeomorphisms obtained as restrictions of generic C2 endomorphisms of an interval with finitely many critical points, all of which are non-flat, and with all periodic points hyperbolic. The ω -limit set of such a restricted endomorphism cannot be infinite, except when the restriction of the endomorphism to the closure of the orbit of some critical point is a minimal homeomorphism of an infinite set.

Multidimensional diffeomorphisms with stable periodic points

2011 ◽  
Vol 84 (3) ◽  
pp. 808-810
Author(s):  
E. V. Vasil’eva
Keyword(s):  
Periodic Points ◽  
Stable Periodic
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