Tubular neighborhoods of orbits of power-logarithmic germs

Abstract

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.