Self-similar \(p\)-adic fractal strings and their complex dimensions. (English) Zbl 1187.28014
Summary: We develop a geometric theory of self-similar \(p\)-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar \(p\)-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar \(p\)-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings.
MSC:
| 28A80 | Fractals |
| 37P20 | Dynamical systems over non-Archimedean local ground fields |
| 11S40 | Zeta functions and \(L\)-functions |
Keywords:
fractal geometry; \(p\)-adic analysis; zeta functions; complex dimensions; self-similarity; lattice strings and Minkowski dimensionReferences:
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