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

Abstract

In this paper, we consider the self-similar measure $\nu_\lambda=\mathrm{law}\left(\sum_{j \geq 0} \xi_j \lambda^j\right)$ on $\mathbb{R}$, where $|\lambda|<1$ and the $\xi_j \sim \nu$ are independent, identically distributed with respect to a measure $\nu$ finitely supported on $\mathbb{Z}$. One example of this is the classical Bernoulli convolution. It is known that for certain combinations of algebraic $\lambda$ and $\nu$ uniform on an interval, $\nu_\lambda$ is absolutely continuous and its Fourier transform has power decay (\cite{garsia1}, \cite{feng}); in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy $h_{\lambda}(\nu)$ is maximal. We show that absolute continuity and power Fourier decay occur when $\lambda$ and $\nu$ are such that $h_{\lambda}(\nu)$ is maximal and classify all combinations for which this is the case. We find that if an algebraic $\lambda$ without a Galois conjugate of modulus exactly one has a $\nu$ such that $h_{\lambda}(\nu)$ is maximal, then all Galois conjugates of $\lambda$ must be smaller in modulus than one and $\nu$ must satisfy a certain finite set of linear equations in terms of $\lambda$.