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Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems

Part of: Classical measure theory Diophantine approximation, transcendental number theory Complex dynamical systems

Published online by Cambridge University Press:  30 June 2020

TUSHAR DAS Open the ORCID record for TUSHAR DAS [Opens in a new window] ,
LIOR FISHMAN ,
DAVID SIMMONS  and
MARIUSZ URBAŃSKI
TUSHAR DAS
Affiliation:
University of Wisconsin–La Crosse, Department of Mathematics & Statistics, 1725 State Street, La Crosse, WI54601, USA (e-mail: tdas@uwlax.edu)
LIOR FISHMAN
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX76203-5017, USA (e-mail: lior.fishman@unt.edu, urbanski@unt.edu)
DAVID SIMMONS
Affiliation:
434 Hanover Ln, Irving, TX75062, USA (e-mail: david9550@gmail.com)
MARIUSZ URBAŃSKI
Affiliation:
University of North Texas, Department of Mathematics, 1155 Union Circle #311430, Denton, TX76203-5017, USA (e-mail: lior.fishman@unt.edu, urbanski@unt.edu)
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Abstract

We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

Keywords

metric Diophantine approximationgeometric measure theoryfractal geometryconformal dynamicsiterated function system (IFS)

MSC classification

Primary: 11J13: Simultaneous homogeneous approximation, linear forms 11J83: Metric theory 28A75: Length, area, volume, other geometric measure theory
Secondary: 37F35: Conformal densities and Hausdorff dimension
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 8 , August 2021 , pp. 2311 - 2348
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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References

Bárány, B.. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fund. Math. 206 (2009), 49–59.Google Scholar
Barreira, L. M., Pesin, Y. B. and Schmeling, J.. Dimension and product structure of hyperbolic measures. Ann. of Math. (2) 149(3) (1999), 755–783.Google Scholar
Bergweiler, W. and Eremenko, A.. Meromorphic functions with linearly distributed values and Julia sets of rational functions. Proc. Amer. Math. Soc. 137(7) (2009), 2329–2333.Google Scholar
Blaya, A. B. and López, V. J.. On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps. Discrete Contin. Dyn. Syst. 32(2) (2012), 433–466.Google Scholar
Bowen, R.. Periodic points and measures for Axiom A diffeomorphisms. Trans. Amer. Math. Soc. 154 (1971), 377–397.Google Scholar
Broderick, R., Fishman, L., Kleinbock, D., Reich, A. and Weiss, B.. The set of badly approximable vectors is strongly C 1 incompressible. Math. Proc. Cambridge Philos. Soc. 153(2) (2012), 319–339.Google Scholar
Broderick, R., Fishman, L. and Simmons, D.. Badly approximable systems of affine forms and incompressibility on fractals. J. Number Theory 133(7) (2013), 2186–2205.Google Scholar
Das, T., Fishman, L., Simmons, D. and Urbański, M.. Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math. (N.S.) 24(3) (2018), 2165–2206.Google Scholar
Das, T. and Simmons, D.. The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result. Invent. Math. 210(1) (2017), 85–134.Google Scholar
Das, T., Simmons, D. and Urbański, M.. Dimension rigidity in conformal structures. Adv. Math. 308 (2017), 1127–1186.Google Scholar
Das, T., Simmons, D. and Urbański, M.. Geometry and Dynamics in Gromov Hyperbolic Metric Spaces: With an Emphasis on Non-Proper Settings (Mathematical Surveys and Monographs, 218). American Mathematical Society, Providence, RI, 2017.Google Scholar
Denker, M. and Urbański, M.. Ergodic theory of equilibrium states for rational maps. Nonlinearity 4(1) (1991), 103–134.Google Scholar
Eremenko, A. and van Strien, S. J.. Rational maps with real multipliers. Trans. Amer. Math. Soc. 363(12) (2011), 6453–6463.Google Scholar
Fishman, L., Simmons, D. and Urbański, M.. Diophantine properties of measures invariant with respect to the Gauss map. J. Anal. Math. 122 (2014), 289–315.Google Scholar
Hofbauer, F.. Generic properties of invariant measures for continuous piecewise monotonic transformations. Monatsh. Math. 106(4) (1988), 301–312.Google Scholar
Hofbauer, F.. Local dimension for piecewise monotonic maps on the interval. Ergod. Th. & Dynam. Sys. 15(6) (1995), 1119–1142.Google Scholar
Inoquio-Renteria, I. and Rivera-Letelier, J.. A characterization of hyperbolic potentials of rational maps. Bull. Braz. Math. Soc. (N.S.) 43(1) (2012), 99–127.Google Scholar
Kleinbock, D. Ya., Lindenstrauss, E. and Weiss, B.. On fractal measures and Diophantine approximation. Selecta Math. 10 (2004), 479–523.Google Scholar
Kleinbock, D. Ya. and Margulis, G. A.. Logarithm laws for flows on homogeneous spaces. Invent. Math. 138(3) (1999), 451–494.Google Scholar
Kleinbock, D. Ya. and Weiss, B.. Badly approximable vectors on fractals. Israel J. Math. 149 (2005), 137–170.Google Scholar
Mauldin, R. D., Szarek, T. and Urbański, M.. Graph directed Markov systems on Hilbert spaces. Math. Proc. Cambridge Philos. Soc. 147 (2009), 455–488.Google Scholar
Mauldin, R. D. and Urbański, M.. Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc. (3) 73(1) (1996), 105–154.Google Scholar
Mauldin, R. D. and Urbański, M.. Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets (Cambridge Tracts in Mathematics, 148). Cambridge University Press, Cambridge, 2003.Google Scholar
Milnor, J. W.. Dynamics in One Complex Variable (Annals of Mathematics Studies, 160), 3rd edn. Princeton University Press, Princeton, NJ, 2006.Google Scholar
Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(3–4) (1976), 241–273.Google Scholar
Paulin, F., Pollicott, M. and Schapira, B.. Equilibrium states in negative curvature. Astérisque 373 (2015), viii+281pp.Google Scholar
Pollicott, M. and Simon, K.. The Hausdorff dimension of 𝜆-expansions with deleted digits. Trans. Amer. Math. Soc. 347(3) (1995), 967–983.Google Scholar
Pollington, A. D. and Velani, S. L.. Metric Diophantine approximation and ‘absolutely friendly’ measures. Selecta Math. 11 (2005), 297–307.Google Scholar
Przytycki, F. and Urbański, M.. Conformal Fractals: Ergodic Theory Methods (London Mathematical Society Lecture Note Series, 371). Cambridge University Press, Cambridge, 2010.Google Scholar
Rivera-Letelier, J.. The maximal entropy measure detects non-uniform hyperbolicity. Math. Res. Lett. 17(5) (2010), 851–866.Google Scholar
Sigmund, K.. Generic properties of invariant measures for Axiom A diffeomorphisms. Invent. Math. 11 (1970), 99–109.Google Scholar
Simmons, D.. Conditional measures and conditional expectation; Rohlin’s disintegration theorem. Discrete Contin. Dyn. Syst. 32(7) (2012), 2565–2582.Google Scholar
Simon, K. and Solomyak, B.. On the dimension of self-similar sets. Fractals 10(1) (2002), 59–65.Google Scholar
Stratmann, B. and Urbański, M.. Diophantine extremality of the Patterson measure. Math. Proc. Cambridge Philos. Soc. 140 (2006), 297–304.Google Scholar
Stratmann, B. O. and Velani, S. L.. The Patterson measure for geometrically finite groups with parabolic elements, new and old. Proc. Lond. Math. Soc. (3) 71(1) (1995), 197–220.Google Scholar
Sullivan, D. P.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171–202.Google Scholar
Sullivan, D. P.. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153(3–4) (1984), 259–277.Google Scholar
Szostakiewicz, M., Urbański, M. and Zdunik, A.. Fine inducing and equilibrium measures for rational functions of the Riemann sphere. Israel J. Math. 210(1) (2015), 399–465.Google Scholar
Urbański, M.. Diophantine approximation and self-conformal measures. J. Number Theory 110 (2005), 219–235.Google Scholar
Urbański, M.. Diophantine approximation for conformal measures of one-dimensional iterated function systems. Compos. Math. 141(4) (2005), 869–886.Google Scholar