Realization of all Dold's congruences with stability

Abstract

Dedicated to professor José M. Montesinos in the occasion of his 65th birthday and to the memory of professor Julián Martínez MSC: 37C25 37B30 54H25 Keywords: Conley index Fixed point index Stable fixed points HomeomorphismsThe main goal of this paper is to prove that for each n > 2, every sequence of integers satisfying Dold's congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving R n -homeomorphism at an isolated stable fixed point.We use Conley index techniques even though stable fixed points are not isolated invariant sets.

Fixed point indices of iterated smooth maps in arbitrary dimension

“…In the final section we consider Babenko and Bogatyi problem and the question of the realization of indices in some narrower classes of smooth maps. In particular we discuss the recent result obtained by Ruiz del Portal and Salazar by Conley index methods in dimension 3 [22].…”

“…Notice that in some of our realizations elliptic sectors are used, so they do not satisfy the assumption (10.1) in such cases. Ruiz del Portal and Salazar considered in [22] that problem in dimension 3 by a use of Conley index methods. These authors gave the list (Proposition 1 in [22]) of examples of R 3 -diffeomorphisms satisfying (10.1) which do not cover all sequences admissible by Theorem 3.1 in dimension 3.…”

“…Ruiz del Portal and Salazar considered in [22] that problem in dimension 3 by a use of Conley index methods. These authors gave the list (Proposition 1 in [22]) of examples of R 3 -diffeomorphisms satisfying (10.1) which do not cover all sequences admissible by Theorem 3.1 in dimension 3. They also conjectured that the list is optimal in the following sense.…”

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

“…If an isolated fixed point p is not isolated as an invariant set, then the sequence of indices of the iterates in general is not bounded, even if the fixed point is Lyapunov stable. More precisely, in [20] it is proved that any sequence satisfying Dold's congruences is realized as the sequence of fixed-point indices of the iterates of an R m -homeomorphism at an isolated and stable fixed point (see also [19]).…”