Indices of the iterates of ℝ³ -homeomorphisms at fixed points which are isolated invariant sets

Abstract

Let U ⊂ ℝ3 be an open set and f : U → f(U) ⊂ ℝ3 be a homeomorphism. Let p ∈ U be a fixed point. It is known that if {p} is not an isolated invariant set, then the sequence of the fixed‐point indices of the iterates of f at p, (i(fn, p))n ⩾ 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(fn, p))n ⩾ 1 is periodic. Conversely, we show that, for any periodic sequence of integers (In)n ⩾ 1 satisfying Dold's necessary congruences, there exists an orientation‐preserving homeomorphism such that i(fn, p) = In for every n ⩾ 1. Finally we also present an application to the study of the local structure of the stable/unstable sets at p.

Generating sequences of Lefschetz numbers of iterates

“…In the proof of the above theorem we will make use of the following algebraic result (Proposition 1 in [16]). Consider the matrix C = M m B.…”

Combinatorial scheme of finding minimal number of periodic points for smooth self-maps of simply connected manifolds

“…Nevertheless, last years brought some important results concerning planar homeomorphism [23], R 3 -homeomorphisms [2,24] and holomorphic maps [25,26,27].…”

Local Fixed Point Indices of Iterations of Planar Maps

“…Theorem 2.4 is based on topological properties of the plane and cannot be generalized for higher dimensions. However, it has been recently shown, by the application of Conley index and topological entropy, that {ind( f n , p)} ∞ n=1 is periodic for R 3 -homeomorphism f for which { p} is an isolated invariant set [15].…”

“…Unfortunately, it is usually difficult to establish the exact form of the indices for a given map. Nevertheless, during the last years the description of indices was given for many important classes of maps such as: planar homeomorphisms [14,16,20,22]; R 3 -homeomorphisms [15,23]; smooth maps [4,9,18,24]; and holomorphic maps [3,6,25,26]. In this paper we give the restrictions for indices of some class of planar maps.…”