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

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.

29 pages, 2 figures, to appear in Journal of Differential Equations

Cited by in corpus (8)

• Invariance of the normalized Minkowski content with respect to the ambient space
• Epsilon-neighborhoods of orbits of parabolic diffeomorphisms and cohomological equations
• Box dimension of unit-time map near nilpotent singularity of planar vector field
• Epsilon-neighborhoods of orbits and formal classification of parabolic diffeomorphisms
• Characteristic box dimension of unit-time map near nilpotent singularity of planar vector field and applications
• Fractal zeta functions of orbits of parabolic diffeomorphisms
• Reading analytic invariants of parabolic diffeomorphisms from their orbits
• Minkowski dimension and slow-fast polynomial Liénard equations near infinity