Fine and coarse multifractal zeta-functions: On the multifractal formalism for multifractal zeta-functions

Abstract: Multifractal analysis refers to the study of the local properties of measures and functions, and consists of two parts: the fine multifractal theory and the coarse multifractal theory. The fine and the coarse theory are linked by a web of conjectures known collectively as the Multifractal Formalism. Very roughly speaking the Multifractal Formalism says that the multifractal spectrum from fine theory equals the Legendre transform of the Renyi dimensions from the coarse theory. Recently {\it fine} multifractal zeta-functions, i.e. multifractal zeta-functions designed to produce detailed information about the fine multifractal theory, have been introduced and investigated. The purpose of this work is to complement and expand this study by introducing and investigating {\it coarse} multifractal zeta-functions, i.e. multifractal zeta-functions designed to produce information about the coarse multifractal theory, and, in particular, to establish a {\it Multifractal Fortmalism for Zeta-Functions} linking fine multifractal zeta-functions and coarse multifractal zeta-functions via the Legendre transform. Several applications are given, including applications to multifractal analysis of graph-directed self-conformal measures and multifractal analysis of ergodic Birkhoff averages of continuous functions on graph-directed self-conformal sets.