×

The dimension spectrum of conformal graph directed Markov systems. (English) Zbl 1421.37013

Summary: 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.

MSC:

37C45 Dimension theory of smooth dynamical systems
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems
28A80 Fractals
11K50 Metric theory of continued fractions
11J70 Continued fractions and generalizations
37B10 Symbolic dynamics
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems

Software:

INTLAB
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Atnip, J.: Non-autonomous conformal graph directed Markov systems. arxiv:1706.09978
[2] Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians, Springer Monographs in Mathematics. Springer, Berlin (2007) · Zbl 1128.43001
[3] Bowen, R.: Hausdorff dimension of quasicircles. Inst. Hautes Études Sci. Publ. Math. 50, 11-25 (1979) · Zbl 0439.30032 · doi:10.1007/BF02684767
[4] Capogna, L., Danielli, D., Pauls, S., Tyson, J.: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, vol. 259. Birkhäuser Verlag, Basel (2007) · Zbl 1138.53003
[5] Chousionis, V., Tyson, J. T., Urbański, M.: Conformal graph directed Markov systems on Carnot groups. Mem. AMS · Zbl 1473.37001
[6] Cusick, T.W.: Hausdorff dimension of sets of continued fractions. Q. J. Math. Oxf. Ser. (2) 41(163), 277-286 (1990) · Zbl 0704.11021 · doi:10.1093/qmath/41.3.277
[7] Dajani, K., Hensley, D., Kraaikamp, C., Masarotto, V.: Arithmetic and ergodic properties of ‘flipped’ continued fraction algorithms. Acta Arith. 153(1), 51-79 (2012) · Zbl 1246.28010 · doi:10.4064/aa153-1-4
[8] Edgar, G.A., Mauldin, D.: Multifractal decompositions of digraph recursive fractals. Proc. Lond. Math. Soc. 3(65), 604-628 (1992) · Zbl 0764.28007 · doi:10.1112/plms/s3-65.3.604
[9] Falk, R.S., Nussbaum, R.D.: A new approach to numerical computation of Hausdorff dimension of iterated function systems: applications to complex continued fractions. Integral Equ. Oper. Theory 90(5), 90-61 (2018) · Zbl 1441.11187 · doi:10.1007/s00020-018-2485-z
[10] Falk, R.S., Nussbaum, R.D.: \[C^mCm\] eigenfunctions of Perron-Frobenius operators and a new approach to numerical computation of Hausdorff dimension: applications in R1. J. Fractal Geom. 5(3), 279-337 (2018) · Zbl 1436.37031 · doi:10.4171/JFG/62
[11] Good, I.J.: The fractional dimensional theory of continued fractions. Proc. Camb. Philos. Soc. 37, 199-228 (1941) · JFM 67.0988.03 · doi:10.1017/S030500410002171X
[12] Ghenciu, A., Munday, S., Roy, M.: The Hausdorff dimension spectrum of conformal graph directed Markov systems and applications to nearest integer continued fractions. J. Number Theory 175, 223-249 (2017) · Zbl 1375.11057 · doi:10.1016/j.jnt.2016.09.002
[13] Heinemann, S., Urbański, M.: Hausdorff dimension estimates for infinite conformal iterated function systems. Nonlinearity 15, 727-734 (2002) · Zbl 1015.37019 · doi:10.1088/0951-7715/15/3/312
[14] Hensley, D.: Continued fraction Cantor sets, Hausdorff dimension, and functional analysis. J. Number Theory 40(3), 336-358 (1992) · Zbl 0745.28005 · doi:10.1016/0022-314X(92)90006-B
[15] Hensley, D.: A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets. J. Number Theory 58(1), 9-45 (1996) · Zbl 0858.11039 · doi:10.1006/jnth.1996.0058
[16] Hensley, D.: Continued Fractions. World Scientific Publishing Co Pte. Ltd., Hackensack (2006) · Zbl 1161.11028 · doi:10.1142/5931
[17] Hensley, D.: Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete Contin. Dyn. Syst. 32(7), 2417-2436 (2012) · Zbl 1287.11099 · doi:10.3934/dcds.2012.32.2417
[18] Hurwitz, A.: Über die Entwicklung complexer Grössen in Kettenbrüche. Acta Math. XI, 187-200 (1888) · JFM 20.0201.01
[19] Hurwitz, J.: Über eine besondere Art der Kettenbruch-Entwicklung complexer Grössen. Dissertation, University of Halle (1895) · JFM 26.0235.01
[20] Jenkinson, O.: On the density of Hausdorff dimensions of bounded type continued fraction sets: the Texan conjecture. Stoch. Dyn. 4(1), 63-76 (2004) · Zbl 1089.28006 · doi:10.1142/S0219493704000900
[21] Jenkinson, O., Pollicott, M.: Computing the dimension of dynamically defined sets: E2 and bounded continued fractions. Ergod. Theory Dyn. Syst. 21, 1429-1445 (2001) · Zbl 0991.28009
[22] Jenkinson, O., Pollicott, M.: Rigorous effective bounds on the Hausdorff dimension of continued fraction Cantor sets: a hundred decimal digits for the dimension of E2, Adv. Math, to appear · Zbl 1386.30039
[23] Kesseböhmer, M., Zhu, S.: Dimension sets for infinite IFSs: the Texan conjecture. J. Number Theory 1(116), 230-246 (2006) · Zbl 1085.37018 · doi:10.1016/j.jnt.2005.04.002
[24] Kesseböhmer, M., Kombrink, S.: Minkowski content and fractal Euler characteristic for conformal graph directed systems. J. Fractal Geom. 2(2), 171-227 (2015) · Zbl 1320.28014 · doi:10.4171/JFG/19
[25] Kesseböhmer, M., Kombrink, S.: Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings. arXiv:1702.02854 · Zbl 1320.28014
[26] Mauldin, D., Urbański, M.: Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. (3) 73(1), 105-154 (1996) · Zbl 0852.28005 · doi:10.1112/plms/s3-73.1.105
[27] Mauldin, D., Urbański, M.: Conformal iterated function systems with applications to the geometry of conformal iterated function systems. Trans. Am. Math. Soc. 351, 4995-5025 (1999) · Zbl 0940.28009 · doi:10.1090/S0002-9947-99-02268-0
[28] Mauldin, D., Urbański, M.: Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge Tracts in Mathematics, vol. 148. Cambridge University Press, Cambridge (2003) · Zbl 1033.37025 · doi:10.1017/CBO9780511543050
[29] Mauldin, D., Williams, S.C.: Hausdorff dimension in graph directed constructions. Trans. Am. Math. Soc. 309, 811-829 (1988) · Zbl 0706.28007 · doi:10.1090/S0002-9947-1988-0961615-4
[30] McMullen, C.T.: Hausdorff dimension and conformal dynamics. Computation of dimension, III. Am. J. Math. 120(4), 691-721 (1998) · Zbl 0953.30026 · doi:10.1353/ajm.1998.0031
[31] Pollicott, M., Urbański, M.: Asymptotic Counting in Conformal Dynamical Systems. arXiv:1704.06896 · Zbl 1482.37002
[32] Priyadarshi, A.: Lower bound on the Hausdorff dimension of a set of complex continued fractions. J. Math. Anal. Appl. 449(1), 91-95 (2017) · Zbl 1362.30003 · doi:10.1016/j.jmaa.2016.12.009
[33] Roy, M.: A new variation of Bowen’s formula for graph directed Markov systems. Discrete Contin. Dyn. Syst. 32(7), 2533-2551 (2012) · Zbl 1262.37018 · doi:10.3934/dcds.2012.32.2533
[34] Rump, Siegfried M., INTLAB — INTerval LABoratory, 77-104 (1999), Dordrecht · Zbl 0949.65046 · doi:10.1007/978-94-017-1247-7_7
[35] Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer. 19, 287-449 (2010) · Zbl 1323.65046 · doi:10.1017/S096249291000005X
[36] Schmidt, A.: Diophantine approximation of complex numbers. Acta Math. 134, 1-85 (1975) · Zbl 0329.10023 · doi:10.1007/BF02392098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.