Exact Hausdorff and packing measures of linear Cantor sets with overlaps
Let $K$ be the attractor of a linear iterated function system (IFS) $S_j(x)=\rho_jx+b_j,$ $j=1,\cdots,m$, on the real line satisfying the generalized finite type condition (whose invariant open set $\mathcal{O}$ is an interval) with an irreducible weighted incidence matrix. This condition was introduced by Lau \& Ngai recently as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of $K$ coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let $\alpha$ be the dimension of $K$. In this paper, we state that \begin{equation*} \mathcal{H}^\alpha(K\cap J)\leq |J|^\alpha \end{equation*} for all intervals $J\subset\overline{\mathcal{O}}$, and \begin{equation*} \mathcal{P}^\alpha(K\cap J)\geq |J|^\alpha \end{equation*} for all intervals $J\subset\overline{\mathcal{O}}$ centered in $K$, where $\mathcal{H}^\alpha$ denotes the $\alpha$-dimensional Hausdorff measure and $\mathcal{P}^\alpha$ denotes the $\alpha$-dimensional packing measure. This result extends a recent work of Olsen where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of $K$. Moreover, using these densities theorems, we describe a scheme for computing $\mathcal{H}^\alpha(K)$ exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing $\mathcal{P}^\alpha(K)$ as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer \& Strichartz and Feng, respectively, and apply to some new classes allowing us to include linear Cantor sets with overlaps.