The dimension spectrum of graph directed Markov systems

TL;DR

This paper investigates the dimension spectrum of conformal graph directed Markov systems, revealing its topological properties, and confirms the full spectrum for systems from complex continued fractions, addressing longstanding conjectures.

Contribution

It provides a comprehensive analysis of the dimension spectrum's size and structure, introduces new phenomena related to the parameter θ, and proves the full spectrum for complex continued fractions.

Findings

Dimension spectrum of infinite conformal IFS is compact and perfect.

New phenomena arise related to the parameter θ in graph directed systems.

The IFS from complex continued fractions has full dimension spectrum.

Abstract

In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions regarding its size and topological structure. As a corollary we obtain that the dimension spectrum of infinite conformal iterated function systems is compact and perfect. On the way we revisit the role of the parameter $\theta$ in graph directed Markov systems and we show that new phenomena arise. We also establish topological pressure estimates for subsystems in the abstract setting of symbolic dynamics with countable alphabets. These estimates play a crucial role in our proofs regarding the dimension spectrum, and they allow us to study Hausdorff dimension asymptotics for subsystems. Finally we narrow our focus to the dimension spectrum of conformal iterated function systems and we prove, among other things, that the iterated function system resulting from the complex continued fractions algorithm has full dimension spectrum. We thus give a positive answer to the Texan conjecture for complex continued fractions.