Tubular neighborhoods of orbits of power-logarithmic germs
Résumé
We consider a class of power-logarithmic germs. We call them parabolic Dulac germs, as they appear as Dulac germs (first-return germs) of hyperbolic polycycles. In view of formal or analytic characterization of such a germ f by fractal properties of several of its orbits, we study the tubular \(\varepsilon \)-neighborhoods of orbits of f with initial points \(x_0\). We denote by \(A_f(x_0,\varepsilon )\) the length of such a tubular \(\varepsilon \)-neighborhood. We show that, even if f is an analytic germ, the function \(\varepsilon \mapsto A_f(x_0,\varepsilon )\) does not have a full asymptotic expansion in \(\varepsilon \) in the scale of powers and (iterated) logarithms. Hence, this partial asymptotic expansion cannot contain necessary information for analytic classification. In order to overcome this problem, we introduce a new notion: the continuous time length of the\(\varepsilon \)-neighborhood\(A^c_f(x_0,\varepsilon )\). We show that this function has a full transasymptotic expansion in \(\varepsilon \) in the power, iterated logarithm scale. Moreover, its asymptotic expansion extends the initial, existing part of the asymptotic expansion of the classical length \(\varepsilon \mapsto A_f(x_0,\varepsilon )\). Finally, we prove that this initial part of the asymptotic expansion determines the class of formal conjugacy of the Dulac germ f.