# American Institute of Mathematical Sciences

April  2011, 30(1): 313-363. doi: 10.3934/dcds.2011.30.313

## Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups

 1 Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka, 560-0043 2 Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430

Received  August 2009 Revised  October 2010 Published  February 2011

We consider the dynamics of semi-hyperbolic semigroups generated by finitely many rational maps on the Riemann sphere. Assuming that the nice open set condition holds it is proved that there exists a geometric measure on the Julia set with exponent $h$ equal to the Hausdorff dimension of the Julia set. Both $h$-dimensional Hausdorff and packing measures are finite and positive on the Julia set and are mutually equivalent with Radon-Nikodym derivatives uniformly separated from zero and infinity. All three fractal dimensions, Hausdorff, packing and box counting are equal. It is also proved that for the canonically associated skew-product map there exists a unique $h$-conformal measure. Furthermore, it is shown that this conformal measure admits a unique Borel probability absolutely continuous invariant (under the skew-product map) measure. In fact these two measures are equivalent, and the invariant measure is metrically exact, hence ergodic.
Citation: Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313
##### References:
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Qiu and Y. Li, Connectedness of Julia sets for a quadratic random dynamical system,, Ergodic Theory Dynam. Systems, 23 (2003), 1807. doi: 10.1017/S0143385703000129. Google Scholar [15] A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I,, Proc. London Math. Soc. (3), 73 (1996), 358. doi: 10.1112/plms/s3-73.2.358. Google Scholar [16] A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups,, Math. Z., 222 (1996), 161. Google Scholar [17] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere,, Ergodic Theory Dynam. Systems, 3 (1983), 351. Google Scholar [18] M. Martens, The existence of σ-finite invariant measures, Applications to real one-dimensional dynamics,, Front for the Math., (). Google Scholar [19] P. Mattila, "Geometry of Sets and Measures in Euclidean spaces. Fractals and Rectifiability,", Cambridge Studies in Advanced Mathematics, 44 (1995). Google Scholar [20] R. D. Mauldin, T. Szarek and M. Urbański, Graph directed Markov systems on Hilbert spaces,, Math. Proc. Cambridge Phil. Soc., 147 (2009), 455. doi: 10.1017/S0305004109002448. Google Scholar [21] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc. (3), 73 (1996), 105. doi: 10.1112/plms/s3-73.1.105. Google Scholar [22] R. 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 [23] J. Milnor, "Dynamics in One Complex Variable (Third Edition),", Annals of Mathematical Studies, (2006). Google Scholar [24] V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry,, preprint 2008, (2008). Google Scholar [25] W. Parry, "Entropy and Generators in Ergodic Theory,", Mathematics Lecture Note Series, (1969). Google Scholar [26] F. Przytycki and M. Urbański, "Fractals in the Plane - The Ergodic Theory Methods,", to be published from Cambridge University Press, (). Google Scholar [27] D. Ruelle, "Thermodynamic Formalism,", Encyclopedia of Math. and Appl., 5 (1978). Google Scholar [28] R. Stankewitz, Completely invariant Julia sets of polynomial semigroups,, Proc. Amer. Math. Soc., 127 (1999), 2889. doi: 10.1090/S0002-9939-99-04857-1. Google Scholar [29] R. Stankewitz, Completely invariant sets of normality for rational semigroups,, Complex Variables Theory Appl., 40 (2000), 199. Google Scholar [30] R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's,, Proc. Amer. Math. Soc., 128 (2000), 2569. doi: 10.1090/S0002-9939-00-05313-2. Google Scholar [31] R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357. Google Scholar [32] R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups,, to appear in Trans. Amer. Math. Soc., (). Google Scholar [33] D. Steinsaltz, Random logistic maps and Lyapunov exponents,, Indag. Mathem., 12 (2001), 557. Google Scholar [34] H. Sumi, On dynamics of hyperbolic rational semigroups,, J. Math. Kyoto Univ., 37 (1997), 717. Google Scholar [35] H. Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups,, Kodai Mathematical Journal, 21 (1998), 10. doi: 10.2996/kmj/1138043831. Google Scholar [36] H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995. doi: 10.1088/0951-7715/13/4/302. Google Scholar [37] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563. Google Scholar [38] H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390. Google Scholar [39] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893. doi: 10.1017/S0143385705000532. Google Scholar [40] H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane,, Appl. Math. Comput., 187 (2007), 489. doi: 10.1016/j.amc.2006.08.149. Google Scholar [41] H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62. Google Scholar [42] H. Sumi, Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets,, Discrete and Continuous Dynamical Systems Ser. A, 29 (2011), 1205. doi: 10.3934/dcds.2011.29.1205. Google Scholar [43] H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, (). Google Scholar [44] H. Sumi, Dynamics of postcritically bounded polynomial semigroups III: Classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles,, Ergodic Theory Dynam. Systems, 30 (2010), 1869. doi: 10.1017/S0143385709000923. Google Scholar [45] H. Sumi, Dynamics of postcritically bounded polynomial semigroups,, preprint 2007, (2007). Google Scholar [46] H. Sumi, Interaction cohomology of forward or backward self-similar systems,, Adv. Math., 222 (2009), 729. doi: 10.1016/j.aim.2009.04.007. Google Scholar [47] H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc., 102 (2011), 50. doi: 10.1112/plms/pdq013. Google Scholar [48] H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics,, preprint 2010, (2010). Google Scholar [49] H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps,, Monatsh. Math., 156 (2009), 371. doi: 10.1007/s00605-008-0016-8. Google Scholar [50] H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups,, Ergodic Theory Dynam. Systems, 30 (2010), 601. doi: 10.1017/S0143385709000297. Google Scholar [51] M. Urbański, Rational functions with no recurrent critical points,, Ergodic Theory Dynam. Systems, 14 (1994), 391. Google Scholar [52] M. Urbański, Geometry and ergodic theory of conformal non-recurrent dynamics,, Ergodic Theory Dynam. Systems, 17 (1997), 1449. doi: 10.1017/S014338579708646X. Google Scholar [53] P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (1982). Google Scholar [54] W. Zhou and and F. Ren, The Julia sets of the random iteration of rational functions,, Chinese Sci. Bulletin, 37 (1992), 969. Google Scholar

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##### References:
 [1] J. Aaronson, "An Introduction to Infinite Ergodic Theory,", Mathematical Surveys and Monographs Vol. \textbf{50}, 50 (1997). Google Scholar [2] R. Brück, Geometric properties of Julia sets of the composition of polynomials of the form $z^2+c_n$,, Pacific J. Math., 198 (2001), 347. doi: 10.2140/pjm.2001.198.347. Google Scholar [3] R. Brück, M. Büger and S. Reitz, Random iterations of polynomials of the form $z^2+c_n$: Connectedness of Julia sets,, Ergodic Theory Dynam. Systems, 19 (1999), 1221. doi: 10.1017/S0143385799141658. Google Scholar [4] M. Büger, Self-similarity of Julia sets of the composition of polynomials,, Ergodic Theory Dynam. Systems, 17 (1997), 1289. doi: 10.1017/S0143385797086458. Google Scholar [5] M. Büger, On the composition of polynomials of the form $z^2+c_n$,, Math. Ann., 310 (1998), 661. Google Scholar [6] L. Carleson, P. W. Jones and J. -C. Yoccoz, Julia and John,, Bol. Soc. Brazil. Math., 25 (1994), 1. Google Scholar [7] M. Denker and M. Urbański, On the existence of conformal measures,, Trans. Amer. Math. Soc., 328 (1991), 563. doi: 10.2307/2001795. Google Scholar [8] M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps,, Nonlinearity, 4 (1991), 103. doi: 10.1088/0951-7715/4/1/008. Google Scholar [9] M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere,, Nonlinearity, 4 (1991), 365. doi: 10.1088/0951-7715/4/2/008. Google Scholar [10] R. Devaney, "An Introduction to Chaotic Dynamical Systems,", Reprint of the second (1989) edition. Studies in Nonlinearity, (1989). Google Scholar [11] K. Falconer, "Techniques in Fractal Geometry,", John Wiley & Sons, (1997). Google Scholar [12] H. Federer, "Geometric Measure Theory,", Springer, (1969). Google Scholar [13] J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynam. Systems, 11 (1991), 687. doi: 10.1017/S0143385700006428. Google Scholar [14] Z. Gong, W. Qiu and Y. Li, Connectedness of Julia sets for a quadratic random dynamical system,, Ergodic Theory Dynam. Systems, 23 (2003), 1807. doi: 10.1017/S0143385703000129. Google Scholar [15] A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I,, Proc. London Math. Soc. (3), 73 (1996), 358. doi: 10.1112/plms/s3-73.2.358. Google Scholar [16] A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups,, Math. Z., 222 (1996), 161. Google Scholar [17] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere,, Ergodic Theory Dynam. Systems, 3 (1983), 351. Google Scholar [18] M. Martens, The existence of σ-finite invariant measures, Applications to real one-dimensional dynamics,, Front for the Math., (). Google Scholar [19] P. Mattila, "Geometry of Sets and Measures in Euclidean spaces. Fractals and Rectifiability,", Cambridge Studies in Advanced Mathematics, 44 (1995). Google Scholar [20] R. D. Mauldin, T. Szarek and M. Urbański, Graph directed Markov systems on Hilbert spaces,, Math. Proc. Cambridge Phil. Soc., 147 (2009), 455. doi: 10.1017/S0305004109002448. Google Scholar [21] R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc. (3), 73 (1996), 105. doi: 10.1112/plms/s3-73.1.105. Google Scholar [22] R. 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 [23] J. Milnor, "Dynamics in One Complex Variable (Third Edition),", Annals of Mathematical Studies, (2006). Google Scholar [24] V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry,, preprint 2008, (2008). Google Scholar [25] W. Parry, "Entropy and Generators in Ergodic Theory,", Mathematics Lecture Note Series, (1969). Google Scholar [26] F. Przytycki and M. Urbański, "Fractals in the Plane - The Ergodic Theory Methods,", to be published from Cambridge University Press, (). Google Scholar [27] D. Ruelle, "Thermodynamic Formalism,", Encyclopedia of Math. and Appl., 5 (1978). Google Scholar [28] R. Stankewitz, Completely invariant Julia sets of polynomial semigroups,, Proc. Amer. Math. Soc., 127 (1999), 2889. doi: 10.1090/S0002-9939-99-04857-1. Google Scholar [29] R. Stankewitz, Completely invariant sets of normality for rational semigroups,, Complex Variables Theory Appl., 40 (2000), 199. Google Scholar [30] R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's,, Proc. Amer. Math. Soc., 128 (2000), 2569. doi: 10.1090/S0002-9939-00-05313-2. Google Scholar [31] R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357. Google Scholar [32] R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups,, to appear in Trans. Amer. Math. Soc., (). Google Scholar [33] D. Steinsaltz, Random logistic maps and Lyapunov exponents,, Indag. Mathem., 12 (2001), 557. Google Scholar [34] H. Sumi, On dynamics of hyperbolic rational semigroups,, J. Math. Kyoto Univ., 37 (1997), 717. Google Scholar [35] H. Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups,, Kodai Mathematical Journal, 21 (1998), 10. doi: 10.2996/kmj/1138043831. Google Scholar [36] H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995. doi: 10.1088/0951-7715/13/4/302. Google Scholar [37] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563. Google Scholar [38] H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390. Google Scholar [39] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893. doi: 10.1017/S0143385705000532. Google Scholar [40] H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane,, Appl. Math. Comput., 187 (2007), 489. doi: 10.1016/j.amc.2006.08.149. Google Scholar [41] H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62. Google Scholar [42] H. Sumi, Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets,, Discrete and Continuous Dynamical Systems Ser. A, 29 (2011), 1205. doi: 10.3934/dcds.2011.29.1205. Google Scholar [43] H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, (). Google Scholar [44] H. Sumi, Dynamics of postcritically bounded polynomial semigroups III: Classification of semi-hyperbolic semigroups and random Julia sets which are Jordan curves but not quasicircles,, Ergodic Theory Dynam. Systems, 30 (2010), 1869. doi: 10.1017/S0143385709000923. Google Scholar [45] H. Sumi, Dynamics of postcritically bounded polynomial semigroups,, preprint 2007, (2007). Google Scholar [46] H. Sumi, Interaction cohomology of forward or backward self-similar systems,, Adv. Math., 222 (2009), 729. doi: 10.1016/j.aim.2009.04.007. Google Scholar [47] H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc., 102 (2011), 50. doi: 10.1112/plms/pdq013. Google Scholar [48] H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics,, preprint 2010, (2010). Google Scholar [49] H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps,, Monatsh. Math., 156 (2009), 371. doi: 10.1007/s00605-008-0016-8. Google Scholar [50] H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups,, Ergodic Theory Dynam. Systems, 30 (2010), 601. doi: 10.1017/S0143385709000297. Google Scholar [51] M. Urbański, Rational functions with no recurrent critical points,, Ergodic Theory Dynam. Systems, 14 (1994), 391. Google Scholar [52] M. Urbański, Geometry and ergodic theory of conformal non-recurrent dynamics,, Ergodic Theory Dynam. Systems, 17 (1997), 1449. doi: 10.1017/S014338579708646X. Google Scholar [53] P. Walters, "An Introduction to Ergodic Theory,", Springer-Verlag, (1982). Google Scholar [54] W. Zhou and and F. Ren, The Julia sets of the random iteration of rational functions,, Chinese Sci. Bulletin, 37 (1992), 969. Google Scholar
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