Continuity of Hausdorff measure for conformal dynamical systems

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October 2013, 33(10): 4647-4692. doi: 10.3934/dcds.2013.33.4647

Continuity of Hausdorff measure for conformal dynamical systems

Tomasz Szarek ^1, , Mariusz Urbański ^2, and Anna Zdunik ^3,

1.	Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk, Poland
2.	Department of Mathematics, University of North Texas, Denton, TX 76203-1430
3.	Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

Received July 2012 Revised January 2013 Published April 2013

Abstract
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Developing the pioneering work of Lars Olsen [14], we deal with the question of continuity of the numerical value of Hausdorff measures of some natural families of conformal dynamical systems endowed with an appropriate natural topology. In particular, we prove such continuity for hyperbolic polynomials from the Mandelbrot set, and more generally for the space of hyperbolic rational functions of a fixed degree. We go beyond hyperbolicity by proving continuity for maps including parabolic rational functions, for example that the parameter $1/4$ is such a continuity point for quadratic polynomials $z\mapsto z^2+c$ for $c\in [0,1/4]$. We prove the continuity of the numerical value of Hausdorff measures also for the spaces of conformal expanding repellers and parabolic ones, more generally for parabolic Walters conformal maps. We also prove some partial continuity results for all conformal Walters maps; these are in general of infinite degree. In order to do this, as one of our tools, we provide a detailed local analysis, uniform with respect to the parameter, of the behavior of conformal maps around parabolic fixed points in any dimension. We also establish continuity of numerical values of Hausdorff measures for some families of infinite $1$-dimensional iterated function systems.

Keywords: Hausdorff measure, conformal dynamical systems., Dimension theory.

Mathematics Subject Classification: 37F35, 37F45, 37F10, 37E0.

Citation: Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete and Continuous Dynamical Systems, 2013, 33 (10) : 4647-4692. doi: 10.3934/dcds.2013.33.4647

References:

[1]	J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibered systems and parabolic rational maps, Transactions of A.M.S., 337 (1993), 495-548. doi: 10.1090/S0002-9947-1993-1107025-2.
[2]	H. Akter and M. Urbański, Real analyticity of hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$, Preprint 2009, to appear Illinois J. Math.
[3]	O. Bodart and M. Zinsmeister, Quelques resultats sur la dimension de Hausdorff des ensembles de Julia des polynomes quadratiques, Fund. Math., 151 (1996), 121-137.
[4]	R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES, 50 (1979), 11-25.
[5]	M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc., 43 (1991), 107-118. doi: 10.1112/jlms/s2-43.1.107.
[6]	M. Denker and M. Urbański, On absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math., 3 (1991), 561-579. doi: 10.1515/form.1991.3.561.
[7]	K. Falconer, "The Geometry of Fractal Sets," Cambridge University Press, 1986.
[8]	H. Federer, "Geometric Measure Theory," Springer, 1969.
[9]	J. Kotus and M. Urbański, Conformal, Geometric and invariant measures for transcendental expanding functions, Math. Annalen., 324 (2002), 619-656. doi: 10.1007/s00208-002-0356-y.
[10]	P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.
[11]	D. Mauldin and M. Urbański, Parabolic iterated function systems, Ergod. Th. & Dynam. Sys., 20 (2000), 1423-1447. doi: 10.1017/S0143385700000778.
[12]	D. Mauldin and M. Urbański, Fractal measures for parabolic IFS, Adv. in Math., 168 (2002), 225-253. doi: 10.1006/aima.2001.2049.
[13]	D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets," Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511543050.
[14]	L. Olsen, Hausdorff and packing measure functions of self- similar sets: continuity and measurability, Ergod. Th. & Dynam. Sys., 28 (2008), 1635-1655. doi: 10.1017/S0143385707000922.
[15]	F. Przytycki and M. Urbański, "Conformal Fractals. Ergodic Theory Methods," London Mathematical Society Lecture Notes Series 371, Cambridge Univ. Press, 2010.
[16]	M. Roy and M. Urbański, Regularity properties of Hausdorff dimension in conformal infinite IFS, Ergodic Th. & Dynam. Sys., 25 (2005), 1961-1983. doi: 10.1017/S0143385705000313.
[17]	M. Roy, H. Sumi and M. Urbański, Lambda-topology vs. pointwise topology, Ergodic Th. and Dynam. Sys., 29 (2009), 685-713. doi: 10.1017/S0143385708080292.
[18]	D. Ruelle, Repellers for real analytic maps, Ergod. Th. & Dynam. Sys., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.
[19]	H. Sumi and M. Urbański, Real analyticity of hausdorff dimension for expanding rational semigroups, Ergod. Th. & Dynam. Sys., 30 (2010), 601-633. doi: 10.1017/S0143385709000297.
[20]	M. Urbański, On Hausdorff dimension of Julia set with a rationally indifferent periodic point, Studia Math., 97 (1991), 167-188.
[21]	M. Urbański, Parabolic Cantor sets, Fund. Math., 151 (1996), 241-277.
[22]	M. Urbański, Analytic families of semihyperbolic generalized polynomial-like mappings, Monatsh. Für Math., 159 (2010), 133-162. doi: 10.1007/s00605-008-0081-z.
[23]	M. Urbański and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergod. Th. & Dynam. Sys., 24 (2004), 279-315. doi: 10.1017/S0143385703000208.
[24]	M. Urbański and A. Zdunik, The parabolic map $f_{1/e}(z)=(1/e)e^z$, Indagationes Math., 13 (2004), 419-433. doi: 10.1016/S0019-3577(04)80009-0.
[25]	M. Urbański and M. Zinsmeister, Geometry of hyperbolic Julia-Lavaurs sets, Indagationes Math., 12 (2001), 273-292. doi: 10.1016/S0019-3577(01)80032-X.
[26]	P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Transactions of A.M.S., 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1.
[27]	A. Zdunik, Parabolic orbifolds and the dimension of maximal measure for rational maps, Invent. Math., 99 (1990), 627-640. doi: 10.1007/BF01234434.

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References:

[1]	J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibered systems and parabolic rational maps, Transactions of A.M.S., 337 (1993), 495-548. doi: 10.1090/S0002-9947-1993-1107025-2.
[2]	H. Akter and M. Urbański, Real analyticity of hausdorff dimension of Julia sets of parabolic polynomials $f_{\lambda}(z)=z(1-z-\lambda z^{2})$, Preprint 2009, to appear Illinois J. Math.
[3]	O. Bodart and M. Zinsmeister, Quelques resultats sur la dimension de Hausdorff des ensembles de Julia des polynomes quadratiques, Fund. Math., 151 (1996), 121-137.
[4]	R. Bowen, Hausdorff dimension of quasi-circles, Publ. Math. IHES, 50 (1979), 11-25.
[5]	M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent periodic point, J. London Math. Soc., 43 (1991), 107-118. doi: 10.1112/jlms/s2-43.1.107.
[6]	M. Denker and M. Urbański, On absolutely continuous invariant measures for expansive rational maps with rationally indifferent periodic points, Forum Math., 3 (1991), 561-579. doi: 10.1515/form.1991.3.561.
[7]	K. Falconer, "The Geometry of Fractal Sets," Cambridge University Press, 1986.
[8]	H. Federer, "Geometric Measure Theory," Springer, 1969.
[9]	J. Kotus and M. Urbański, Conformal, Geometric and invariant measures for transcendental expanding functions, Math. Annalen., 324 (2002), 619-656. doi: 10.1007/s00208-002-0356-y.
[10]	P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.
[11]	D. Mauldin and M. Urbański, Parabolic iterated function systems, Ergod. Th. & Dynam. Sys., 20 (2000), 1423-1447. doi: 10.1017/S0143385700000778.
[12]	D. Mauldin and M. Urbański, Fractal measures for parabolic IFS, Adv. in Math., 168 (2002), 225-253. doi: 10.1006/aima.2001.2049.
[13]	D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets," Cambridge Univ. Press, 2003. doi: 10.1017/CBO9780511543050.
[14]	L. Olsen, Hausdorff and packing measure functions of self- similar sets: continuity and measurability, Ergod. Th. & Dynam. Sys., 28 (2008), 1635-1655. doi: 10.1017/S0143385707000922.
[15]	F. Przytycki and M. Urbański, "Conformal Fractals. Ergodic Theory Methods," London Mathematical Society Lecture Notes Series 371, Cambridge Univ. Press, 2010.
[16]	M. Roy and M. Urbański, Regularity properties of Hausdorff dimension in conformal infinite IFS, Ergodic Th. & Dynam. Sys., 25 (2005), 1961-1983. doi: 10.1017/S0143385705000313.
[17]	M. Roy, H. Sumi and M. Urbański, Lambda-topology vs. pointwise topology, Ergodic Th. and Dynam. Sys., 29 (2009), 685-713. doi: 10.1017/S0143385708080292.
[18]	D. Ruelle, Repellers for real analytic maps, Ergod. Th. & Dynam. Sys., 2 (1982), 99-107. doi: 10.1017/S0143385700009603.
[19]	H. Sumi and M. Urbański, Real analyticity of hausdorff dimension for expanding rational semigroups, Ergod. Th. & Dynam. Sys., 30 (2010), 601-633. doi: 10.1017/S0143385709000297.
[20]	M. Urbański, On Hausdorff dimension of Julia set with a rationally indifferent periodic point, Studia Math., 97 (1991), 167-188.
[21]	M. Urbański, Parabolic Cantor sets, Fund. Math., 151 (1996), 241-277.
[22]	M. Urbański, Analytic families of semihyperbolic generalized polynomial-like mappings, Monatsh. Für Math., 159 (2010), 133-162. doi: 10.1007/s00605-008-0081-z.
[23]	M. Urbański and A. Zdunik, Real analyticity of Hausdorff dimension of finer Julia sets of exponential family, Ergod. Th. & Dynam. Sys., 24 (2004), 279-315. doi: 10.1017/S0143385703000208.
[24]	M. Urbański and A. Zdunik, The parabolic map $f_{1/e}(z)=(1/e)e^z$, Indagationes Math., 13 (2004), 419-433. doi: 10.1016/S0019-3577(04)80009-0.
[25]	M. Urbański and M. Zinsmeister, Geometry of hyperbolic Julia-Lavaurs sets, Indagationes Math., 12 (2001), 273-292. doi: 10.1016/S0019-3577(01)80032-X.
[26]	P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Transactions of A.M.S., 236 (1978), 121-153. doi: 10.1090/S0002-9947-1978-0466493-1.
[27]	A. Zdunik, Parabolic orbifolds and the dimension of maximal measure for rational maps, Invent. Math., 99 (1990), 627-640. doi: 10.1007/BF01234434.

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