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A notion of Euler characteristic for fractals. (English) Zbl 1118.28006
Let \(F\) be a compact set, and \(F_\varepsilon\) be its \(\varepsilon\)-neighborhood. The authors introduce the notions of the fractal Euler number
\[ \chi_f(F):=\lim_{\varepsilon\to 0} (\varepsilon/b)^\sigma \chi(F_\varepsilon) \]
and the average fractal Euler number
\[ \chi_f^a(F):=\lim_{\varepsilon\to 0} \frac{1}{\log \delta} \int_0^\delta (\varepsilon/b)^\sigma \chi(F_\varepsilon)\frac{d\varepsilon}{\varepsilon}, \] provided \(\chi(F_\varepsilon)\) is determined for all \(\varepsilon>0\). Here
\[ \sigma=\sigma(F) :=\inf\{ t\geq 0:\,\, \varepsilon^t | \chi(F_\varepsilon)| \text{ is bounded as }\varepsilon\to 0\} \] is the Euler exponent of \(F\). These notions do not possess the same properties as the Euler characteristics of a cell complex, but they allow to distinguish fractal sets with the same fractal dimension, but different (average) fractal Euler number. The authors give many examples in which it is possibly to calculate explicitly these numbers. In the special case of self-similar parallel sets, and under the assumption of the existence of \(\chi(F_\varepsilon)\), the authors find sufficient conditions for the existence of (average) limits (Theorem 2.1). Their proof shows a beautiful interplay between probability theory and geometry – to prove the existence of limits, one of the versions of the renewal theorem is applied to so-called overlap function:
\[ R(\varepsilon)=\chi(F_\varepsilon) -\sum_{i=1}^N \chi((S_iF)_\varepsilon), \] under the additional assumption that there exist constants \(c,\gamma>0\) such that \(| R(\varepsilon)| \leq c\varepsilon^{\gamma-s}\) for all \(0<\varepsilon\leq 1.\) This condition is essential for applying the renewal theorem. Further, the authors discuss the case when the \(\varepsilon\)-neighborhood of a set is in a convex ring, and formulate conditions, under which assumptions of Theorem 2.1 are satisfied. The paper contains many examples, which illustrate the introduced notions.

28A80 Fractals
60K05 Renewal theory
28A75 Length, area, volume, other geometric measure theory
Full Text: DOI
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