Multifractal Analysis of Inhomogeneous Multinomial Measures with Non-Doubling Projections

Measures were constructed on symbolic spaces that satisfy an extended multifractal formalism, where Olsen’s functions [Formula: see text] and [Formula: see text] differ, and their Legendre transforms have the expected interpretation in terms of dimensions. These measures were composed with a Gray code and projected onto the unit interval to obtain doubling measures. It was demonstrated that the projected measure retains the same Olsen’s functions as the original and also satisfies the extended multifractal formalism. In this paper, we show that the use of a Gray code is not essential to achieve these results, even when dealing with non-doubling measures. Moreover, general results on multifractal analysis of inhomogeneous multinomial measures with their non-doubling projections are obtained. The key points of the proof include two main components: the study of weak doubling properties and the method of constructing auxiliary measure to get sharp bound for the dimension under consideration.