  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.
