Reading multiplicity in unfoldings from ε-neighborhoods of orbits

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 ε-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 ε-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 ε-neighborhoods of orbits.