Extended multifractal formalism of some non-doubling measures

TL;DR

This paper extends the multifractal formalism to certain non-doubling measures, demonstrating that Gray code construction is unnecessary for these results, and confirms the formalism's applicability beyond doubling measures.

Contribution

It proves that the extended multifractal formalism applies to non-doubling measures without using Gray code constructions, broadening the scope of the formalism.

Findings

Extended multifractal formalism applies to non-doubling measures.

Gray code is not necessary for the formalism to hold.

Projected measures retain the same Olsen's functions as original measures.

Abstract

In a previous work \cite{She} we constructed measures on symbolic spaces which satisfy an extended multifractal formalism (in the sense that Olsen's functions $b$ and $B$ differ and that their Legendre transforms have the expected interpretation in terms of dimensions). These measures are composed with a Gray code and projected onto the unit interval so to get doubling measures. Then we were able to show that the projected measure has the same Olsen's functions as the one it comes from and that it also fulfills the extended multifractal formalism. Here we show that the use of a Gray code is not necessary to get these results, although dealing with non doubling measures.