Indices of the iterates of $R^3$-homeomorphisms at Lyapunov stable fixed points

TL;DR

This paper constructs specific three-dimensional homeomorphisms with a Lyapunov stable fixed point at the origin, demonstrating that the sequence of fixed point indices can grow arbitrarily fast, highlighting differences from planar cases.

Contribution

It introduces a method to create 3D homeomorphisms with prescribed index growth at a Lyapunov stable fixed point, revealing new behaviors not seen in planar homeomorphisms.

Findings

Sequences of fixed point indices can grow arbitrarily fast in 3D homeomorphisms.

Strong differences exist between 3D and planar fixed point index behaviors.

Constructed examples show unbounded index growth at Lyapunov stable points.

Abstract

Given any positive sequence (\{c_n\}_{n \in {\Bbb N}}), we construct orientation preserving homeomorphisms (f:{\Bbb R}^3 \to {\Bbb R}^3) such that (Fix(f)=Per(f)=\{0\}), (0) is Lyapunov stable and (\limsup \frac{|i(f^m, 0)|}{c_m}= \infty). We will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms.