The dimension spectrum of conformal graph directed Markov systems

• Vasileios Chousionis ^[1] ; Dmitriy Leykekhman ^[1] ; Mariusz Urbanski ^[2]
  1. [1] University of Connecticut

     University of Connecticut

     Town of Mansfield, Estados Unidos

     
  2. [2] University of North Texas

     University of North Texas

     Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 3, 2019
• Idioma: inglés
• DOI: 10.1007/s00029-019-0487-6
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• Resumen

  • 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 prove that the dimension spectrum of infinite conformal iterated function systems is compact and perfect. On the way we revisit the role of the parameter θ 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. As a result, we provide a positive answer to the Texan conjecture for complex continued fractions.