Covering number on inhomogeneous graph-directed self-similar sets

TL;DR

This paper investigates the asymptotic behavior of the covering number of inhomogeneous graph-directed self-similar sets, linking it to Minkowski dimension and the structure of the contraction group, revealing conditions for convergence and divergence.

Contribution

It characterizes the asymptotic covering number behavior of inhomogeneous graph-directed self-similar sets, extending understanding of their geometric complexity.

Findings

The asymptotic behavior depends on the Minkowski dimension of the attractor.

Convergence of the scaled covering number occurs under an integrability condition.

The limit form varies with the nature of the log-contraction group (real line or lattice).

Abstract

For a strongly connected inhomogeneous graph-directed self-similar set $K^C$ satisfying the strong open set condition, we characterize the asymptotic behaviour of the $r$-covering number $N_r(K^C)$ as $r \downarrow 0$ in terms of the Minkowski dimension $s_0(G)$ of the attractor. If $\int_0^\infty e^{-s_0(G)t}N_{e^{-t}}(C_i)\,\mathrm{d} t<\infty$ for all vertices $i$, then $e^{-s_0(G)t}N_{e^{-t}}(K^C)$ has a limit as $t\to\infty$, which is a positive constant when the log-contraction group $G_M$ is $\mathbb{R}$ and a positive periodic function when $G_M$ is a lattice; if the integral diverges for some $i$, the limit is infinite.