On absolute continuity and maximal Garsia entropy for self-similar measures with algebraic contraction ratio

We consider the self-similar measure ν_λ=law(∑j ≥ 0 ξⱼ λʲ) on R , where |λ| < 1 and the ξⱼ ~ ν are independent, identically distributed with respect to a measure ν finitely supported on Z . One example of such a measure is a Bernoulli convolution. It is known that for certain combinations of algebraic λ and ν uniform on an interval, ν_λ is absolutely continuous and its Fourier transform has power decay; in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy hλ(ν) is maximal. In this paper, we show that the phenomenon of hλ(ν) being maximal is equivalent to absolute continuity of a self-affine measure μ_λ , which is naturally associated to λ and projects onto ν_λ . We also classify all combinations for which this phenomenon occurs: we find that if an algebraic λ without a Galois conjugate of modulus exactly one has a ν such that hλ(ν) is maximal, then all Galois conjugates of λ must be smaller in modulus than one and ν must satisfy a certain finite set of linear equations in terms of λ . Lastly, we show that in this case, the measure μ_λ is not only absolutely continuous but also has power Fourier decay, which implies the same for ν_λ .