Indices of the iterates of R3R^3R3-homeomorphisms at Lyapunov stable fixed points (original) (raw)

Indices of the iterates of -homeomorphisms at Lyapunov stable fixed points

Journal of Differential Equations, 2008

Given any positive sequence {c n } n∈N , we construct orientation preserving homeomorphisms f : R 3 → R 3 such that F ix(f) = P er(f) = {0}, 0 is Lyapunov stable and lim sup |i(f m ,0)| cm = ∞. We will use our results to discuss and to point out some strong dierences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms.

Indices of the iterates of R3homeomorphisms at Lyapunov stable xed points

2013

Given any positive sequence {cn}n∈N, we construct orientation preserving homeomorphisms f: R 3 → R 3 such that F ix(f) = P er(f) = {0}, 0 is Lyapunov stable and lim sup |i(f m,0)| = ∞. We will use our cm results to discuss and to point out some strong di erences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms. 1. Introduction. The computation of the sequence of the indices, or the sequence of Lefschetz numbers, of the iterates of a map is an important and non-trivial problem. When a xed point is an isolated invariant set of an orientation preserving

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

2009

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).

Index 1 fixed points of orientation reversing planar homeomorphisms

Topological Methods in Nonlinear Analysis, 2015

Let U ⊂ R 2 be an open subset, f : U → f (U) ⊂ R 2 be an orientation reversing homeomorphism and let 0 ∈ U be an isolated, as a periodic orbit, fixed point. The main theorem of this paper says that if the fixed point indices i R 2 (f, 0) = i R 2 (f 2 , 0) = 1 then there exists an orientation preserving dissipative homeomorphism ϕ : R 2 → R 2 such that f 2 = ϕ in a small neighbourhood of 0 and {0} is a global attractor for ϕ. As a corollary we have that for orientation reversing planar homeomorphisms a fixed point, which is an isolated fixed point for f 2 , is asymptotically stable if and only if it is stable. We also present an application to periodic differential equations with symmetries where orientation reversing homeomorphisms appear naturally.

A Poincaré formula for the fixed point indices of the iterates of arbitrary planar homeomorphisms

2009

Let U ⊂ R 2 be an open subset and f : U → R 2 be an arbitrary local homeomorphism with F ix(f ) = {p}. We compute the fixed point indices of the iterates of f at p, i R 2 (f k , p), and we identify these indices in dynamical terms. Therefore we obtain a sort of Poincaré index formula without differentiability assumptions. Our techniques apply equally to both, orientation preserving and orientation reversing homeomorphisms. We present some new results, specially in the orientation reversing case.

The geometric index and attractors of homeomorphisms of

Ergodic Theory and Dynamical Systems, 2021

In this paper we focus on compacta$K \subseteq \mathbb {R}^3$which possess a neighbourhood basis that consists of nested solid tori$T_i$. We call these sets toroidal. Making use of the classical notion of the geometric index of a curve inside a torus, we introduce the self-geometric index of a toroidal setK, which roughly captures how each torus$T_{i+1}$winds inside the previous$T_i$as$i \rightarrow +\infty .Wethenusethisindextoobtainsomeresultsabouttherealizabilityoftoroidalsetsasattractorsforhomeomorphismsof. We then use this index to obtain some results about the realizability of toroidal sets as attractors for homeomorphisms of.Wethenusethisindextoobtainsomeresultsabouttherealizabilityoftoroidalsetsasattractorsforhomeomorphismsof\mathbb {R}^3$.

The geometric index and attractors of homeomorphisms of mathbbR3\mathbb{R}^3mathbbR3

arXiv (Cornell University), 2019

In this paper we focus on compacta K ⊆ R 3 which possess a neighbourhood basis that consists of nested solid tori T i. We call these sets toroidal. In [2] we defined the genus of a toroidal set as a generalization of the classical notion of genus from knot theory. Here we introduce the self-geometric index of a toroidal set K, which captures how each torus T i+1 winds inside the previous T i. We use this index in conjunction with the genus to approach the problem of whether a toroidal set can be realized as an attractor for a flow or a homeomorphism of R 3. We obtain a complete characterization of those that can be realized as attractors for flows and exhibit uncountable families of toroidal sets that cannot be realized as attractors for homeomorphisms.

Local Fixed Point Indices of Iterations of Planar Maps

Journal of Dynamics and Differential Equations, 2011

Let f : U → R 2 be a continuous map, where U is an open subset of R 2 . We consider a fixed point p of f which is neither a sink nor a source and such that {p} is an isolated invariant set. Under these assumption we prove, using Conley index methods and Nielsen theory, that the sequence of fixed point indices of iterations {ind(f n , p)} ∞ n=1 is periodic, bounded by 1, and has infinitely many non-positive terms, which is a generalization of Le Calvez and Yoccoz theorem [Annals of Math., 146 (1997), 241-293] onto the class of non-injective maps. We apply our result to study the dynamics of continuous maps on 2-dimensional sphere.

Fixed point index of iterations of local homeomorphisms of the plane: a Conley index approach

Topology, 2002

Let U be an open subset of R 2 and let f : U → R 2 be a local homeomorphism. Let p ∈ U be a non-repeller ÿxed point of f such that {p} is an isolated invariant set. We introduce a particular class of index pairs for {p} that we call generalized ÿltration pairs. The computation of the ÿxed point index of any iteration of f at p is quite easy once one knows a generalized ÿltration pair. The existence of generalized ÿltration pairs provides a short and elementary proof of a theorem of P. Le Calvez and J.C. Yoccoz (Ann. of Meth. 146 (1997) 241-293), and it also allows to compute the ÿxed point index of any iteration of arbitrary local homeomorphisms. ?