arXiv subjects

Patrice Le Calvez

Publications and source records attributed to Patrice Le Calvez.

22 records · Page 2Linked to original sources

arXiv2009

Indices of the iterates of ({\Bbb R}^3)-homeomorphisms at fixed points which are isolated invariant sets

Let (U \subset {\mathbb R}^3) be an open set and (f:U \to f(U) \subset {\mathbb R}^3) be a homeomorphism. Let (p \in U) be a fixed point. It is known that, if (\{p\}) is not an isolated invariant set, the sequence of the fixed point indices of the iterates of (f) at (p), ((i(f^n,p))_{n\geq 1}), is, in general, unbounded. The main goal of this paper is to show that when (\{p\}) is an isolated invariant set, the sequence ((i(f^n,p))_{n\geq 1}) is periodic. Conversely, we show that for any periodic sequence of integers ((I_n)_{n \geq1}) satisfying Dold's necessary congruences, there exists an orientation preserving homeomorphism such that (i(f^n,p)=I_n) for every (n\geq 1). Finally we also present an application to the study of the local structure of the stable/unstable sets at (p).

math.DS↗

arXiv2009

Une nouvelle preuve du theoreme de point fixe de Handel

M Handel has proved in [Topology 38 (1999) 235--264] a fixed point theorem for an orientation preserving homeomorphism of the open unit disk, that may be extended to the closed disk and that satisfies a linking property of orbits. We give here a new proof of Handel's fixed point theorem, based on Brouwer theory and some plane topology arguments. We will slightly improve the theorem by proving the existence of a simple closed curve of index 1. This index result was known to be true under an additional hypothesis and has been used by different authors (J Franks [NYJM 2 (1996) 1--19, Trans.AMS 348 (1996) 2637--2662] S Matsumoto [Topol. Appl. 104 (2000) 191--214]) to study homeomorphisms of surfaces.

math.GT↗

arXiv2000

Regions of instability for non-twist maps

In this paper we consider an analog of the regions of instability for twist maps in the context of area preserving diffeomorphisms which are not twist maps. Several properties analogous to those of classical regions of instability are proved.

math.DS↗

arXiv1997

Stably non-synchronizable maps of the plane

Pecora and Carroll presented a notion of synchronization where an (n-1)-dimensional nonautonomous system is constructed from a given $n$-dimensional dynamical system by imposing the evolution of one coordinate. They noticed that the resulting dynamics may be contracting even if the original dynamics are not. It is easy to construct flows or maps such that no coordinate has synchronizing properties, but this cannot be done in an open set of linear maps or flows in $\R^n$, $n\geq 2$. In this paper we give examples of real analytic homeomorphisms of $\R^2$ such that the non-synchronizability is stable in the sense that in a full $C^0$ neighborhood of the given map, no homeomorphism is synchronizable.

math.DS↗