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# Tubular neighborhoods of orbits of power-logarithmic germs

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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.
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-02384780
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Soumis le : jeudi 28 novembre 2019 - 15:07:08
Dernière modification le : vendredi 25 novembre 2022 - 10:12:06

### Citation

Pavao Mardešić, M. Resman, Jean-Philippe Rolin, V. Županović. Tubular neighborhoods of orbits of power-logarithmic germs. Journal of Dynamics and Differential Equations, 2021, 33, pp.395-443. ⟨10.1007/s10884-019-09812-8⟩. ⟨hal-02384780⟩

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