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Porosity in Conformal Dynamical Systems

Part of: Set functions and measures on spaces with additional structure Classical measure theory Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  14 April 2021

VASILEIOS CHOUSIONIS  and
MARIUSZ URBAŃSKI
VASILEIOS CHOUSIONIS
Affiliation:
Department of Mathematics, University of Connecticut, 341 Marsfield Road 41009, Stars, CT 06269-3009, U.S.A. e-mail: vasileios.chousionis@uconn.edu
MARIUSZ URBAŃSKI
Affiliation:
Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, 7X76203-5017, U.S.A. e-mail: mariuszurbanski@unt.edu
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Abstract

In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that their limit sets are porous in large (in the sense of category and dimension) subsets. We also provide natural geometric and dynamic conditions under which the limit set of a GDMS is upper porous or mean porous. On the other hand, we prove that if the limit set of a GDMS is not porous, then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction.

We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterisation of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are uniformly upper porous and mean porous almost everywhere.

We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

MSC classification

Primary: 28A75: Length, area, volume, other geometric measure theory
Secondary: 28C10: Set functions and measures on topological groups or semigroups, Haar measures, invariant measures 35R03: Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc.
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 2 , March 2022 , pp. 303 - 371
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported in part by the Simons Foundation Collaboration grant no. 521845.

‡

Supported in part by the Simons Foundation Collaboration grant no. 581668.

References

Beliaev, D. and Smirnov, S.. On dimension of porous measures. Math. Ann. 323 (2002), no. 1, 123–141.CrossRefGoogle Scholar
Beliaev, D., Järvenpää, E., Järvenpää, M., Käenmäki, A., Rajala, T., Smirnov, S. and Suomala, V.. Packing dimension of mean porous measures. J. Lond. Math. Soc. (2) 80 (2009), no. 2, 514–530.CrossRefGoogle Scholar
Bergweiler, W.. Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29 (1993), 151–188.CrossRefGoogle Scholar
Bruna, J.. Boundary interpolation sets for holomorphic functions smooth to the boundary and BMO. Trans. Amer. Math. Soc. 264 (1981), 393–409.CrossRefGoogle Scholar
Chousionis, V.. Directed porosity on conformal iterated function systems and weak convergence of singular integrals. Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 1, 215–232.Google Scholar
Chousionis, V., Leykekhman, D. and Urbański, M.. The dimension spectrum of graph directed Markov systems. Selecta Math. 25 (2019), no. 3.CrossRefGoogle Scholar
Chousionis, V., Tyson, J. T. and Urbański, M.. Conformal graph directed Markov systems on Carnot groups. Mem. Amer. Math. Soc. 266 (2020), no. 1291.Google Scholar
Conway, J. B.. Functions of One Complex Variable I (Springer, 1978).CrossRefGoogle Scholar
Denjoy, A.. Sur une propriété des séries trigonométriques. Verlag v.d.G.V. der Wisen Natuur (Afd. 1920).Google Scholar
Dobbs, N.. Nice sets and invariant densities in complex dynamics. Ergodic Theory Dynam. System 211 (1992), 73–86.Google Scholar
Dolženko, E.. Boundary properties of arbitrary functions (in Russian). Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3–14.Google Scholar
Conrad, K.. The Gaussian Integers. (2016). Available at http://www.math.uconn.edu/˜kconrad/blurbs/ugradnumthy/Zinotes.pdf.Google Scholar
Eckmann, J. P., Järvenpää, E. and Järvenpää, E.. Porosities and dimensions of measures. Nonlinearity 13 (2000), 1–18.CrossRefGoogle Scholar
Edgar, G. A. and Mauldin, D.. Multifractal decompositions of digraph recursive fractals. Proc. London Math. Soc. 3 (1992), no. 65, 604–628.CrossRefGoogle Scholar
Falk, R. S. and Nussbaum, R. D.. A new approach to numerical computation of Hausdorff dimen- sion of iterated function systems: applications to complex continued fractions. Integral Equations Operator Theory 90 (2018), no. 5, 90–61.CrossRefGoogle Scholar
Gethner, E., Wagon, S. and Wick, B.. A stroll through the Gaussian primes. American Mathematical Monthly 105 (1998).Google Scholar
Granlund, S., Linqvist, P. and Martio, O.. F-harmonic measure in space. Ann. Acad. Sci. Fenn. Math. 7 (1982), 233–247.CrossRefGoogle Scholar
Hensley, D.. Continued fractions (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006).CrossRefGoogle Scholar
Hlawka, E., Schoissengeier, J. and Taschner, R.. Geometric and analytic number theory. Universitext (Springer-Verlag, Berlin, 1991). Translated from the 1986 German edition by Charles Thomas.CrossRefGoogle Scholar
Hawkins, J. and Koss, L.. Parametrised dynamics of the Weiestrass elliptic function. Monatsh. Math. 137 (2002), 273–301.CrossRefGoogle Scholar
Hurwitz, A.. Über die Entwicklung complexer Grössen in Kettenbrüche. Acta Math. XI (1888), 187–200.CrossRefGoogle Scholar
Hurwitz, J., Über eine besondere Art der Kettenbruch–Entwicklung complexer Grössen. Dissertation. University of Halle (1895).Google Scholar
Järvenpää, E., Järvenpää, M. and Mauldin, D.. Deterministic and random aspects of porosities. Discrete Contin. Dyn. Syst. 8 (2002), no. 1, 121–136.CrossRefGoogle Scholar
Käenmäki, A. and Suomala, V.. Conical upper density theorems and porosity of measures. Adv. Math. 217 (2008), 952–966.CrossRefGoogle Scholar
Käenmäki, A. and Suomala, V.. Nonsymmetric conical upper density and k-porosity. Trans. Amer. Math. Soc. 363 (2011), no. 3, 1183–1195.CrossRefGoogle Scholar
Kesseböhmer, M. and Kombrink, S.. Minkowski content and fractal Euler characteristic for conformal graph directed systems. J. Fractal Geom. 2 (2015), no. 2, 171–227.CrossRefGoogle Scholar
Kesseböhmer, M. and Kombrink, S.. Minkowski measurability of infinite conformal graph directed systems and application to Apollonian packings, ArXiv 1702.02854.Google Scholar
Koskela, P. and Rohde, S.. Hausdorff dimension and mean porosity. Math. Ann. 309 (1997), no. 4, 593–609.CrossRefGoogle Scholar
Kotus, J. and Urbański, M.. Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions. Bull. London Math. Soc. 35 (2003), 269–275.CrossRefGoogle Scholar
Kotus, J. and Urbański, M.. Geometry and ergodic theory of non-recurrent elliptic functions. J. d’Analyse Math. 93 (2004), 35–102.CrossRefGoogle Scholar
Kotus, J. and Urbański, M.. Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Ellip- tic Functions, ArXiv 2007.13235, 2020.Google Scholar
Lindenstrass, J., Preiss, D. and Tišer, J.. Fréchet differentiability of Lipschitz maps and porous sets in Banach spaces, Princeton University Press, 2012.Google Scholar
Martio, O. and Vuorinen, M.. Whitney cubes, p-capacity, and Minkowski content. Exposition. Math. 5 (1987), 17–40.Google Scholar
Mattila, P.. Distribution of sets and measures along planes. J. London Math. Soc. 38 (1988), 125–132.CrossRefGoogle Scholar
Mauldin, D. and Urbański, M.. Dimensions and measures in infinite iterated function systems. Proc. London Math. Soc. (3) 73 (1996), no. 1, 105–154.CrossRefGoogle Scholar
Mauldin, D. and Urbański, M.. Graph directed Markov systems, Geometry and dynamics of limit sets, Cambridge Tracts in Mathematics, vol. 148, Cambridge University Press, Cambridge, 2003.CrossRefGoogle Scholar
Mauldin, D. and Williams, S. C.. Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309 (1988), 811–829.CrossRefGoogle Scholar
Mayer, V. and Urbański, M.. Geometric Thermodynamical Formalism and Real Analyticity for Mero- morphic Functions of Finite Order. Ergod. Th. & Dynam. Sys. 28 (2008), 915–946.CrossRefGoogle Scholar
Mayer, V. and Urbański, M.. Thermodynamical Formalism and Multifractal Analysis for Meromorphic Functions of Finite Order, Memoirs of AMS 203 (2010).CrossRefGoogle Scholar
Mayer, V. and Urbański, M.. Gibbs and equilibrium measures for elliptic functions. Math. Zeit. 250 (2005), 657–683.CrossRefGoogle Scholar
Nieminen, T.. Generalized mean porosity and dimension. Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 143–172.Google Scholar
Pollicott, M. and Urbański, M.. Asymptotic Counting in Conformal Dynamical Systems. Mem. Amer. Math. Soc. to appear.Google Scholar
Preiss, D. and Zajček, L.. Sigma-porous sets in products of metric spaces and sigma-directionally porous sets in Banach spaces. Real Anal. Exchange 24 (1998/99), no. 1, 295–313.CrossRefGoogle Scholar
Preiss, D. and Zajček, L.. Directional derivatives of Lipschitz functions. Israel J. Math. 125 (2001), 1–27.CrossRefGoogle Scholar
Priyadarshi, A.. Lower bound on the Hausdorff dimension of a set of complex continued fractions. J. Math. Anal. Appl. 449 (2017), no. 1, 91–95.CrossRefGoogle Scholar
Przytycki, F. and Rivera-Letelier, J.. Statistical properties of topological Collet-Eckmann maps. Ann. Sci. École Norm. Sup. 40 (2007), 135–178.CrossRefGoogle Scholar
Przytycki, F. and Rohde, S.. Porosity of Collet-Eckmann Julia sets, Fundamenta Mathematicae 155 (1998), no. 2, 189–199.Google Scholar
Przytycki, F. and Urbański, M.. Rigidity of tame rational functions. Bull. Pol. Acad. Sci., Math. 47 (1999), no. 2, 163–182.Google Scholar
Przytycki, F. and Urbański, M.. Conformal Fractals, Ergodic Theory Methods, Cambridge University Press, 2017.Google Scholar
Rivera-Letelier, J.. A connecting lemma for rational maps satisfying a no–growth condition. Ergod. Th. & Dynam. Sys. 27 (2007), 595–636.CrossRefGoogle Scholar
Roy, M.. A new variation of Bowen’s formula for graph directed Markov systems. Discrete Contin. Dyn. Syst. 32 (2012), no. 7, 2533–2551.CrossRefGoogle Scholar
Salli, A.. On the Minkowski dimension of strongly porous fractal sets in R n . Proc. London Math. Soc. 62 (1991), no. 7, 353–372.CrossRefGoogle Scholar
Shmerkin, P.. The dimension of weakly mean porous measures: a probabilistic approach. Int. Math. Res. Not. IMRN 9 (2012), 2010–2033, DOI 10.1093/imrn/rnr085.Google Scholar
Skorulski, B. and Urbański, M.. Finer fractal geometry for analytic families of conformal dynamical systems. Dyn. Sys. 29 (2014), 369–398.CrossRefGoogle Scholar
Speight, G.. Porosity and differentiability. (ProQuest LLC, Ann Arbor, MI, 2013). Ph.D thesis– University of Warwick.Google Scholar
Stillwell, J.. Elements of number theory. Undergraduate Texts in Math. (Springer–Verlag, New York, 2003).Google Scholar
Väisälä, J.. Porous sets and quasisymmetric maps. Trans. Amer. Math. Soc. 299 (1987), 525–533.CrossRefGoogle Scholar
Urbański, M.. Porosity in Conformal Infinite Iterated Function Systems. J. Number Theory 88 (2001), no. 2, 283–312.CrossRefGoogle Scholar
Zajíček, L.. Porosity and σ-porosity. Real Anal. Exchange 13 (1987/88), 314–350.CrossRefGoogle Scholar