Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen's $b$ and $B$ functions

TL;DR

This paper investigates the multifractal properties of inhomogeneous multinomial measures, revealing that their Olsen's functions are analytic with tangent graphs, and provides dimension-based interpretations of their Legendre transforms.

Contribution

It introduces a new class of inhomogeneous multinomial measures with analytic Olsen's functions and explores their multifractal spectra and geometric properties.

Findings

Olsen's functions $b$ and $B$ are analytic for these measures.

The graphs of $b$ and $B$ differ except at two tangent points.

Legendre transforms relate to the measures' dimensions.

Abstract

Inhomogeneous multinomial measures on the mixed symbolic spaces and the real line are given. By counting the zeros of the corresponding generalized Dirichlet polynomials, one obtains a probability measure whose Olsen's functions $b$ and $B$ are analytic and their graphs differ except at two points where they are tangent. Also, interpretations of the Legendre transform of $b$ and $B$ are given in terms of dimensions.