2023/03/14 by Lauritz Streck, Streck, Lauritz · 4 citations
Mathematics · Computer Science · Physics and Astronomy · #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #Advanced Thermodynamics and Statistical Mechanics
paper · pdf · doi:10.48550/arxiv.2303.07785
In this paper, we consider the self-similar measure νλ=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 this is the classical Bernoulli convolution. It is known that for certain combinations of algebraic λ and ν uniform on an interval, νλ is absolutely continuous and its Fourier transform has power decay (\citegarsia1, \citefeng); in the proof, it is exploited that for these combinations, a quantity called the Garsia entropy hλ(ν) is maximal. We show that absolute continuity and power Fourier decay occur when λ and ν are such that hλ(ν) is maximal and classify all combinations for which this is the case. 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.