Covering number on inhomogeneous graph-directed self-similar sets

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}){\text{d}}t<\infty for all vertices i , then e^{-s_0(G)t}N_{e^{-t}}(K^{C}) has a limit 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.