Multiplicity of fixed points and growth of epsilon-neighbourhoods of orbits

TL;DR

This paper extends the understanding of the relationship between fixed point multiplicity and epsilon-neighborhood growth of orbits to non-differentiable functions using a new critical Minkowski order, with applications in dynamical systems.

Contribution

It introduces a new notion of critical Minkowski order to generalize fixed point multiplicity and epsilon-neighborhood relationships to non-differentiable functions.

Findings

Established the relationship between multiplicity and epsilon-neighborhood growth in non-differentiable cases.

Introduced the concept of critical Minkowski order for analyzing orbit neighborhoods.

Applied the theory to Poincare maps, homoclinic loops, and Abelian integrals.

Abstract

We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on epsilon of the length of epsilon-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered before (Elezovic, Zubrinic, Zupanovic) in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity of fixed points and the dependence on epsilon of the length of epsilon-neighborhoods of orbits in non-differentiable cases. Applications include in particular Poincare maps near homoclinic loops and hyperbolic 2-cycles, and Abelian integrals. This is a new approach to estimate the cyclicity, by computing the length of the epsilon-neighborhood of one orbit of the Poincare map (for example numerically), and by comparing it to the appropriate scale.