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

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

2011-08-23

arxiv.org/abs/1108.4707v2

Mathematics

Dynamical Systems

25 pages, 2 figures

Scientific paper

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 by 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. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With these new definitions, 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 map near homoclinic loop and Abelian integrals.

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