Multifractal spectra and multifractal zeta-functions

Abstract

We introduce multifractal zeta-functions providing precise information of a very general class of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions. More precisely, we prove that these and more general multifractal spectra equal the abscissae of convergence of the associated zetafunctions.2000 Mathematics Subject Classification. Primary: 28A78. Secondary: 37D30, 37A45., but to find explicit expressions for the quantities M µ,δ (q) and N µ,δ (α; r) themselves. The purpose of this work can bee seen as a first step in this direction. Again, summarizing this somewhat more succinctly, the present work is concentrated on the following problem: Present work:This work explores methods of finding explicit expressions for M µ,δ (q) and N µ,δ (α; r) . It is clear that finding explicit expressions for M µ,δ (q) and N µ,δ (α; r) is a more challenging undertaking than determining the limiting behaviour of the ratios log M µ,δ (q) − log δ and log N µ,δ (α;r) − log δ ; indeed, if explicit expressions for M µ,δ (q) and N µ,δ (α; r) are known, then the limiting behaviour of the ratios log M µ,δ (q) − log δ and log N µ,δ (α;r) − log δ

Dynamical multifractal zeta-functions and fine multifractal spectra of graph-directed self-conformal constructions

“…The key idea in [LapRo,LapLe-VeRo] is both simple and attractive: while traditional zeta-functions are defined by "summing over all data", the multifractal zeta-functions in [LapRo,LapLe-VeRo] are defined by only "summing over data that are multifractally relevant". This idea is also the leitmotif in this work (as well as in earlier work [Bak,MiOl,Ol4]), see, in particular, the first remark following the definition of the zeta-function ζ dyn,U C (ϕ; ·) in Section 4. Ideas similar to those in [LapRo,LapLe-VeRo] have very recently been revisited and investigated in [Bak,MiOl] where the authors introduce related geometric multifractal zeta-functions tailored to study the multifractal spectra of self-conformal measures and a number of connections with very general types of multifractal spectra were established.…”

“…In addition to the distinctively geometric approaches in [Bak,MiOl,Ol2,Ol3], it has been a major challenge to introduce and develop a natural and meaningful theory of dynamical multifractal zeta-functions paralleling the existing powerful theory of dynamical zeta-functions introduced and developed by Ruelle [Rue1,Rue2] and others, see, for example, the surveys and books [Bal1,Bal2,ParPo1,ParPo2] and the references therein. In particular, in the setting of self-conformal constructions, [Ol4] introduced a family of dynamical multifractal zeta-functions designed to provide precise information of very general classes of multifractal spectra, including, for example, the multifractal spectra of self-conformal measures and the multifractal spectra of ergodic Birkhoff averages of continuous functions.…”