Covering number on inhomogeneous graph-directed self-similar sets

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.