Hausdorff and box dimension of self-affine set in non-Archimedean field

TL;DR

This paper extends the theory of fractal dimensions to self-affine sets in non-Archimedean fields, providing a generic formula for their Hausdorff and box dimensions without restrictive assumptions on linear parts.

Contribution

It develops the singular value composition theory in non-Archimedean fields and computes fractal dimensions of self-affine sets, generalizing Falconer's results to this setting.

Findings

Derived formulas for Hausdorff and box dimensions in non-Archimedean fields.

No need for additional assumptions on linear transformation norms.

Results applicable to a broad class of self-affine sets.

Abstract

In this paper we consider affine iterated function systems in locally compact non-Archimedean field $\mathbb{F}$. We establish the theory of singular value composition in $\mathbb{F}$ and compute box and Hausdorff dimension of self-affine set in $\mathbb{F}^n$, in generic sense, which is an analogy of Falconer's result for real case. The result has the advantage that no additional assumptions needed to be imposed on the norms of linear parts of affine transformation while such norms are strictly less than $\frac{1}{2}$ for real case, which benefits from the non-Archimedean metric on $\mathbb{F}$.