Indices of fixed points not accumulated by periodic points

TL;DR

This paper constructs continuous maps on Euclidean spaces with a single fixed point and prescribed fixed point indices sequence, demonstrating the realization of certain algebraic relations in dynamical systems.

Contribution

It proves the existence of maps with a unique fixed point and a specified sequence of fixed point indices satisfying Dold relations, expanding understanding of fixed point index realizability.

Findings

Existence of maps with a single fixed point and prescribed index sequence

Construction of maps satisfying Dold relations

Demonstration of index sequence realizability in Euclidean spaces

Abstract

We prove that for every integer sequence $I$ satisfying Dold relations there exists a map $f : \mathbb{R}^d \to \mathbb{R}^d$, $d \ge 2$, such that $\mathrm{Per(f)} = \mathrm{Fix(f)} = \{o\}$, where $o$ denotes the origin, and $(i(f^n, o))_n = I$.