Conformal Graph Directed Markov Systems on
                    Carnot Groups

Chousionis, Vasilis; Tyson, Jeremy T.; Urbański, Mariusz

doi:10.1090/memo/1291

Cited by 10 publications

(46 citation statements)

References 20 publications

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“…As mentioned earlier the theory of graph directed Markov systems has been recently extended to the sub-Riemannian setting of Carnot groups in [5]. We record that the content of Sections 3-6 is valid for Euclidean as well as Carnot graph directed Markov systems.…”

Section: Introductionmentioning

confidence: 86%

“…Mauldin and the last named author developed a fully fledged theory of Euclidean conformal GDMS with a countable alphabet in [26] stemming from [24]. In the recent monograph [5], Tyson together with the first and last named authors extended the theory of conformal GDMS in the setting of nilpotent stratified Lie groups (Carnot groups) equipped with a sub-Riemannian metric. See also [1,22,23,29,31] for recent advances on various aspects of GDMSs.…”

Section: Introductionmentioning

confidence: 99%

“…In [5,24,26], and in other relevant works, thermodynamic formalism is heavily used in order to study the limit sets of conformal graph directed Markov systems. In particular one needs to study the topological pressure function of the system, which will be defined in Section 3.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The dimension spectrum of conformal graph directed Markov systems

Chousionis

Leykekhman

Urbański

2019

Sel. Math. New Ser.

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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 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 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. We thus give a positive answer to the Texan conjecture for complex continued fractions.

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Section: Introductionmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

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The dimension spectrum of conformal graph directed Markov systems

Chousionis

Leykekhman

Urbański

2019

Sel. Math. New Ser.

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“…Hausdorff dimensions of self-similar and self-affine sets in the Heisenberg setting have been studied, for example, in [1,5,6,8]. In [2,3,16,18], the authors study the Hausdorff dimensions of projections and slices in Heisenberg groups.…”

Section: Introductionmentioning

confidence: 99%

Hausdorff dimension of limsup sets of isotropic rectangles in Heisenberg groups

Myllyoja¹

2021

Preprint

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“…In the Heisenberg group Hausdorff dimensions of self-similar and self-affine sets have been studied e.g. in [1,5,6,11]. Even though the class of affine iterated function systems is quite restrictive -every such system is a horizontal lift of an affine iterated function system on the plane -the dimension calculations involve some subtleties.…”

Section: Introductionmentioning

confidence: 99%