Dimension of homogeneous rational self-similar measures with overlaps

We develop a tool to analyse some self-similar measures with overlaps, those obtained from systems of homotheties with centres in a lattice where the contraction ratios are all equal to the inverse of a natural number L. We obtain the local dimension of the measure as the Shannon entropy of an associated hidden Markov chain divided by the logarithm of L. This result turns out to be useful in the study of the absolute continuity or singularity of the measure, and provides two sequences converging to the dimension of the measure, one of them non-increasing and the other non-decreasing, which allows us to obtain estimates of the dimension.