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It is known that the heuristic principle, referred to as the multifractal formalism, need not hold for self-similar measures with overlap, such as the 3-fold convolution of the Cantor measure and certain Bernoulli convolutions. In this paper we study an important function in the multifractal theory, the L q -spectrum, τ (q), for measures of finite type, a class of self-similar measures that includes these examples. Corresponding to each measure, we introduce finitely many variants on the L q -spectrum which arise naturally from the finite type structure and are often easier to understand than τ . We show that τ is always bounded by the minimum of these variants and is equal to the minimum variant for q ≥ 0. This particular variant coincides with the L q -spectrum of the measure µ restricted to appropriate subsets of its support. If the IFS satisfies particular structural properties, which do hold for the above examples, then τ is shown to be the minimum of these variants for all q. Under certain assumptions on the local dimensions of µ, we prove that the minimum variant for q ≪ 0 coincides with the straight line having slope equal to the maximum local dimension of µ. Again, this is the case with the examples above. More generally, bounds are given for τ and its variants in terms of notions closely related to the local dimensions of µ.

By a Cantor-like measure we mean the unique self-similar probability measure µ satisfyingfor integers 2 ≤ d < m ≤ 2d − 1 and probabilities p i > 0, p i = 1. In the uniform case (p i = 1/m for all i) we show how one can compute the entropy and Hausdorff dimension to arbitrary precision. In the non-uniform case we find bounds on the entropy.

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