# American Institute of Mathematical Sciences

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

## Continuity of Hausdorff measure for conformal dynamical systems

 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

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.
Citation: Tomasz Szarek, Mariusz Urbański, Anna Zdunik. Continuity of Hausdorff measure for conformal dynamical systems. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar [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, (2009).   Google Scholar [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.   Google Scholar [4] R. Bowen, Hausdorff dimension of quasi-circles,, Publ. Math. IHES, 50 (1979), 11.   Google Scholar [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.  doi: 10.1112/jlms/s2-43.1.107.  Google Scholar [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.  doi: 10.1515/form.1991.3.561.  Google Scholar [7] K. Falconer, "The Geometry of Fractal Sets,", Cambridge University Press, (1986).   Google Scholar [8] H. Federer, "Geometric Measure Theory,", Springer, (1969).   Google Scholar [9] J. Kotus and M. Urbański, Conformal, Geometric and invariant measures for transcendental expanding functions,, Math. Annalen., 324 (2002), 619.  doi: 10.1007/s00208-002-0356-y.  Google Scholar [10] P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,", Cambridge Studies in Advanced Mathematics, 44 (1995).   Google Scholar [11] D. Mauldin and M. Urbański, Parabolic iterated function systems,, Ergod. Th. & Dynam. Sys., 20 (2000), 1423.  doi: 10.1017/S0143385700000778.  Google Scholar [12] D. Mauldin and M. Urbański, Fractal measures for parabolic IFS,, Adv. in Math., 168 (2002), 225.  doi: 10.1006/aima.2001.2049.  Google Scholar [13] D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,", Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511543050.  Google Scholar [14] L. Olsen, Hausdorff and packing measure functions of self- similar sets: continuity and measurability,, Ergod. Th. & Dynam. Sys., 28 (2008), 1635.  doi: 10.1017/S0143385707000922.  Google Scholar [15] F. Przytycki and M. Urbański, "Conformal Fractals. Ergodic Theory Methods,", London Mathematical Society Lecture Notes Series 371, (2010).   Google Scholar [16] M. Roy and M. Urbański, Regularity properties of Hausdorff dimension in conformal infinite IFS,, Ergodic Th. & Dynam. Sys., 25 (2005), 1961.  doi: 10.1017/S0143385705000313.  Google Scholar [17] M. Roy, H. Sumi and M. Urbański, Lambda-topology vs. pointwise topology,, Ergodic Th. and Dynam. Sys., 29 (2009), 685.  doi: 10.1017/S0143385708080292.  Google Scholar [18] D. Ruelle, Repellers for real analytic maps,, Ergod. Th. & Dynam. Sys., 2 (1982), 99.  doi: 10.1017/S0143385700009603.  Google Scholar [19] H. Sumi and M. Urbański, Real analyticity of hausdorff dimension for expanding rational semigroups,, Ergod. Th. & Dynam. Sys., 30 (2010), 601.  doi: 10.1017/S0143385709000297.  Google Scholar [20] M. Urbański, On Hausdorff dimension of Julia set with a rationally indifferent periodic point,, Studia Math., 97 (1991), 167.   Google Scholar [21] M. Urbański, Parabolic Cantor sets,, Fund. Math., 151 (1996), 241.   Google Scholar [22] M. Urbański, Analytic families of semihyperbolic generalized polynomial-like mappings,, Monatsh. Für Math., 159 (2010), 133.  doi: 10.1007/s00605-008-0081-z.  Google Scholar [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.  doi: 10.1017/S0143385703000208.  Google Scholar [24] M. Urbański and A. Zdunik, The parabolic map $f_{1/e}(z)=(1/e)e^z$,, Indagationes Math., 13 (2004), 419.  doi: 10.1016/S0019-3577(04)80009-0.  Google Scholar [25] M. Urbański and M. Zinsmeister, Geometry of hyperbolic Julia-Lavaurs sets,, Indagationes Math., 12 (2001), 273.  doi: 10.1016/S0019-3577(01)80032-X.  Google Scholar [26] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances,, Transactions of A.M.S., 236 (1978), 121.  doi: 10.1090/S0002-9947-1978-0466493-1.  Google Scholar [27] A. Zdunik, Parabolic orbifolds and the dimension of maximal measure for rational maps,, Invent. Math., 99 (1990), 627.  doi: 10.1007/BF01234434.  Google Scholar

show all references

##### 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.  doi: 10.1090/S0002-9947-1993-1107025-2.  Google Scholar [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, (2009).   Google Scholar [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.   Google Scholar [4] R. Bowen, Hausdorff dimension of quasi-circles,, Publ. Math. IHES, 50 (1979), 11.   Google Scholar [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.  doi: 10.1112/jlms/s2-43.1.107.  Google Scholar [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.  doi: 10.1515/form.1991.3.561.  Google Scholar [7] K. Falconer, "The Geometry of Fractal Sets,", Cambridge University Press, (1986).   Google Scholar [8] H. Federer, "Geometric Measure Theory,", Springer, (1969).   Google Scholar [9] J. Kotus and M. Urbański, Conformal, Geometric and invariant measures for transcendental expanding functions,, Math. Annalen., 324 (2002), 619.  doi: 10.1007/s00208-002-0356-y.  Google Scholar [10] P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability,", Cambridge Studies in Advanced Mathematics, 44 (1995).   Google Scholar [11] D. Mauldin and M. Urbański, Parabolic iterated function systems,, Ergod. Th. & Dynam. Sys., 20 (2000), 1423.  doi: 10.1017/S0143385700000778.  Google Scholar [12] D. Mauldin and M. Urbański, Fractal measures for parabolic IFS,, Adv. in Math., 168 (2002), 225.  doi: 10.1006/aima.2001.2049.  Google Scholar [13] D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,", Cambridge Univ. Press, (2003).  doi: 10.1017/CBO9780511543050.  Google Scholar [14] L. Olsen, Hausdorff and packing measure functions of self- similar sets: continuity and measurability,, Ergod. Th. & Dynam. Sys., 28 (2008), 1635.  doi: 10.1017/S0143385707000922.  Google Scholar [15] F. Przytycki and M. Urbański, "Conformal Fractals. Ergodic Theory Methods,", London Mathematical Society Lecture Notes Series 371, (2010).   Google Scholar [16] M. Roy and M. Urbański, Regularity properties of Hausdorff dimension in conformal infinite IFS,, Ergodic Th. & Dynam. Sys., 25 (2005), 1961.  doi: 10.1017/S0143385705000313.  Google Scholar [17] M. Roy, H. Sumi and M. Urbański, Lambda-topology vs. pointwise topology,, Ergodic Th. and Dynam. Sys., 29 (2009), 685.  doi: 10.1017/S0143385708080292.  Google Scholar [18] D. Ruelle, Repellers for real analytic maps,, Ergod. Th. & Dynam. Sys., 2 (1982), 99.  doi: 10.1017/S0143385700009603.  Google Scholar [19] H. Sumi and M. Urbański, Real analyticity of hausdorff dimension for expanding rational semigroups,, Ergod. Th. & Dynam. Sys., 30 (2010), 601.  doi: 10.1017/S0143385709000297.  Google Scholar [20] M. Urbański, On Hausdorff dimension of Julia set with a rationally indifferent periodic point,, Studia Math., 97 (1991), 167.   Google Scholar [21] M. Urbański, Parabolic Cantor sets,, Fund. Math., 151 (1996), 241.   Google Scholar [22] M. Urbański, Analytic families of semihyperbolic generalized polynomial-like mappings,, Monatsh. Für Math., 159 (2010), 133.  doi: 10.1007/s00605-008-0081-z.  Google Scholar [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.  doi: 10.1017/S0143385703000208.  Google Scholar [24] M. Urbański and A. Zdunik, The parabolic map $f_{1/e}(z)=(1/e)e^z$,, Indagationes Math., 13 (2004), 419.  doi: 10.1016/S0019-3577(04)80009-0.  Google Scholar [25] M. Urbański and M. Zinsmeister, Geometry of hyperbolic Julia-Lavaurs sets,, Indagationes Math., 12 (2001), 273.  doi: 10.1016/S0019-3577(01)80032-X.  Google Scholar [26] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances,, Transactions of A.M.S., 236 (1978), 121.  doi: 10.1090/S0002-9947-1978-0466493-1.  Google Scholar [27] A. Zdunik, Parabolic orbifolds and the dimension of maximal measure for rational maps,, Invent. Math., 99 (1990), 627.  doi: 10.1007/BF01234434.  Google Scholar
 [1] Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060 [2] Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118. [3] Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017 [4] Markus Böhm, Björn Schmalfuss. Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3115-3138. doi: 10.3934/dcdsb.2018303 [5] Nuno Luzia. On the uniqueness of an ergodic measure of full dimension for non-conformal repellers. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5763-5780. doi: 10.3934/dcds.2017250 [6] Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006 [7] Nhan-Phu Chung. Gromov-Hausdorff distances for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6179-6200. doi: 10.3934/dcds.2020275 [8] Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235 [9] Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785 [10] Richard Sharp. Conformal Markov systems, Patterson-Sullivan measure on limit sets and spectral triples. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2711-2727. doi: 10.3934/dcds.2016.36.2711 [11] Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288 [12] Juan Wang, Yongluo Cao, Yun Zhao. Dimension estimates in non-conformal setting. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3847-3873. doi: 10.3934/dcds.2014.34.3847 [13] Yuan-Ling Ye. Non-uniformly expanding dynamical systems: Multi-dimension. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2511-2553. doi: 10.3934/dcds.2019106 [14] Lana Horvat Dmitrović. Box dimension and bifurcations of one-dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1287-1307. doi: 10.3934/dcds.2012.32.1287 [15] Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591 [16] Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405 [17] Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503 [18] Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457 [19] Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293 [20] Luis Barreira. Dimension theory of flows: A survey. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345

2019 Impact Factor: 1.338