| Abstract: |
| We study fractal zeta functions of orbits of a model tangent to the identity germs on the real line. The set of complex dimensions of a given (fractal) set is defined as the set of poles of the associated Lapidus zeta function. It generalizes the Minkowski dimension and is closely related to the asymptotics of the lengths of the epsilon neighborhood of the given set, i.e., its so-called tube function. In general, the tube function of a given orbit will not have an asymptotic expansion at 0 beyond a certain term due to oscillations generated by the discrete critical time. Nevertheless, we show that a full asymptotic expansion exists in the sense of Schwarz distributions and that this fact is connected to the meromorphicity of the associated Lapidus zeta function in all of \mathbb{C}.
We also relate the above mentioned distributional asymptotic expansion to the associated continuous epsilon neighborhood which is defined by embedding the diffeomorphism into a flow as its time-one map. This is of interest since it is shown that the continuous epsilon neighborhood possesses a full asymptotic expansion at 0. This is a joint work with Maja Resman, University of Zagreb, and Pavao Mardesic, Universite de Bourgogne. |
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