Regularity of multifractal spectra of conformal iterated function systems

TL;DR

This paper studies how the multifractal spectra of infinite conformal iterated function systems depend continuously on the system and introduces a convergence notion that allows extending finite system results to infinite systems.

Contribution

It introduces the concept of regular convergence for cIFS, enabling the transfer of multifractal analysis results from finite to infinite systems without regularity assumptions.

Findings

Established an Exhausting Principle for infinite cIFS

Proved continuity of multifractal spectra under regular convergence

Connected the results to the λ-topology of Roy and Urbański

Abstract

We investigate multifractal regularity for infinite conformal iterated function systems (cIFS). That is we determine to what extent the multifractal spectrum depends continuously on the cIFS and its thermodynamic potential. For this we introduce the notion of regular convergence for families of cIFS not necessarily sharing the same index set, which guarantees the convergence of the multifractal spectra on the interior of their domain. In particular, we obtain an Exhausting Principle for infinite cIFS allowing us to carry over results for finite to infinite systems, and in this way to establish a multifractal analysis without the usual regularity conditions. Finally, we discuss the connections to the $\lambda$-topology introduced by Roy and Urbas{\'n}ki.