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The dimension spectrum of conformal graph directed Markov systems

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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 \(\theta \) 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. As a result, we provide a positive answer to the Texan conjecture for complex continued fractions.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, USA

    Vasileios Chousionis & Dmitriy Leykekhman

  2. Department of Mathematics, University of North Texas, Denton, USA

    Mariusz Urbański

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  1. Vasileios Chousionis
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  2. Dmitriy Leykekhman
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  3. Mariusz Urbański
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Correspondence to Vasileios Chousionis.

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Vasileios Chousionis was supported in part by the Simons Foundation Collaboration Grant No. 521845. Dmitriy Leykekhman was supported in part by the NSF Grant No. DMS-1522555. Mariusz Urbański was supported in part by the NSF Grant No. DMS-1361677.

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Chousionis, V., Leykekhman, D. & Urbański, M. The dimension spectrum of conformal graph directed Markov systems. Sel. Math. New Ser. 25, 40 (2019). https://doi.org/10.1007/s00029-019-0487-6

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