Non-doubling of self-similar measures for all Pisot numbers β in (1,2) except the golden ratio

Determine whether, for every Pisot number β in the interval (1, 2) with β not equal to the golden ratio (i.e., β ≠ (1+√5)/2), the self-similar measure μ_p on [0,1] defined by μ_p = p_1 μ_p ∘ S_1^{-1} + p_2 μ_p ∘ S_2^{-1} with S_1(x) = x/β and S_2(x) = x/β + (1 − 1/β), is non-doubling for all probability weights p = (p_1, p_2) satisfying p_1, p_2 > 0 and p_1 + p_2 = 1.

Background

The paper studies when self-similar measures associated with the iterated function system {S_1(x)=x/β, S_2(x)=x/β+(1−1/β)} are doubling, focusing on the case where overlaps occur. For β in (1,2) satisfying β^m = ∑_{j=0}^{m-1} β^j (a Pisot number), the authors prove that the measure μ_p is always non-doubling for m ≥ 3, while for m = 2 (the golden ratio) Yung’s result shows that μ_p is doubling if and only if p_1 = p_2 = 1/2.

Motivated by these results and using a method that provides an algorithmic check for Pisot β, the authors conjecture a broader non-doubling phenomenon: for any Pisot β in (1,2) other than the golden ratio, the associated self-similar measures μ_p should always be non-doubling, regardless of the weight p.