engleski

Tubular neighborhoods of orbits of power- logarithmic germs

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 ε-neighborhoods of orbits of f with initial points x0. We denote by A f (x0, ε) the length of such a tubular ε- neighborhood.We show that, even if f is an analytic germ, the function ε → A f (x0, ε) does not have a full asymptotic expansion in ε 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 ε-neighborhood Ac f (x0, ε). We show that this function has a full transasymptotic expansion in ε in the power, iterated logarithm scale. Moreover, its asymptotic expansion extends the initial, existing part of the asymptotic expansion of the classical length ε → A f (x0, ε). Finally, we prove that this initial part of the asymptotic expansion determines the class of formal conjugacy of the Dulac germ f .

Dulac map ; Fractal properties of orbits ; ε-Neighborhoods ; Power-logarithm asymptotic expansions ; Transseries ; Formal and analytic invariants ; Embedding in a flow

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