2021/04/14 by Rutar, Alex
#28A78 (Primary) 28A80 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2104.06997
We study self-similar measures in ℝ satisfying the weak separation condition along with weak technical assumptions which are satisfied in all known examples. For such a measure μ, we show that there is a finite set of concave functions \τ1,…,τm\ such that the Lq-spectrum of μ is given by min\τ1,…,τm\ and the multifractal spectrum of μ is given by max\τ1^*,…,τm^*\, where τi^* denotes the concave conjugate of τi. In particular, the measure μ satisfies the multifractal formalism if and only if its multifractal spectrum is a concave function. This implies that μ satisfies the multifractal formalism at values corresponding to points of differentiability of the Lq-spectrum. We also verify existence of the limit for the Lq-spectra of such measures for every q∈ℝ. As a direct application, we obtain many new results and simple proofs of well-known results in the multifractal analysis of self-similar measures satisfying the weak separation condition.