Reading multiplicity in unfoldings from epsilon-neighborhoods of orbits

Abstract

We consider generic 1-parameter unfoldings of parabolic vector fields. It is known that the box dimension of orbits of their time-one maps is discontinuous at the bifurcation value. Here, we expand asymptotically the Lebesgue measure of the epsilon-neighborhoods of orbits of the time-one maps in a Chebyshev scale, uniformly with respect to the bifurcation parameter. We use the so-called Ecalle-Roussarie-type compensators. We read from the expansion the number of hyperbolic points born in the unfolding of the parabolic point (i.e. the codimension of the bifurcation). 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. In order 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.