Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 61, pp. 1-19.

Title: Reading multiplicity in unfoldings from epsilon-neighborhoods of orbits

Authors: Renato Huzak (Hasselt Univ., Diepenbeek, Belgium)
         Pavao Mardesic (Univ. Bourgogne Europe, France)
         Maja Resman (Univ. of Zagreb, Croatia)
         Vesna Zupanovic (Univ. of Zagreb, Croatia)

Abstract:
We consider generic  analytic 1-parameter unfoldings of saddle-node
germs of analytic vector fields on the real line, their time-one maps and the
Lebesgue measure of $\varepsilon$-neighborhoods of the orbits of these time-one maps.
The box dimension of an orbit gives the asymptotics of the principal term of this
Lebesgue measure and it is known that it is discontinuous at bifurcation parameters.
To recover continuous dependence of the asymptotics on the parameter,
here we expand asymptotically the Lebesgue measure of $\varepsilon$-neighborhoods of
orbits of time-one maps in a Chebyshev system, uniformly with respect to the
bifurcation parameter.
We use Ecalle-Roussarie-type compensators.
We show how the number of fixed points of the time-one map born in the universal
analytic unfolding of the parabolic point
corresponds to the number of  terms vanishing in
this uniform expansion of the Lebesgue measure of $\varepsilon$-neighborhoods of orbits. 

Submitted February 21, 2025. Published June 10, 2025.
Math Subject Classifications: 37G10, 34C23, 28A80, 37C45, 37M20.
Key Words: Unfoldings; epsilon-neighborhoods; compensators; Chebyshev scale.