# American Institute of Mathematical Sciences

July  2012, 32(7): 2591-2606. doi: 10.3934/dcds.2012.32.2591

## Bowen parameter and Hausdorff dimension for expanding 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, Denton, TX 76203-1430

Received  November 2009 Revised  August 2010 Published  March 2012

We estimate the Bowen parameters and the Hausdorff dimensions of the Julia sets of expanding finitely generated rational semigroups. We show that the Bowen parameter is larger than or equal to the ratio of the entropy of the skew product map $\tilde{f}$ and the Lyapunov exponent of $\tilde{f}$ with respect to the maximal entropy measure for $\tilde{f}$. Moreover, we show that the equality holds if and only if the generators are simultaneously conjugate to the form $a_{j}z^{\pm d}$ by a M\"{o}bius transformation. Furthermore, we show that there are plenty of expanding finitely generated rational semigroups such that the Bowen parameter is strictly larger than $2$.
Citation: 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
##### References:
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Comput., 187 (2007), 489.  doi: 10.1016/j.amc.2006.08.149.  Google Scholar [24] H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62.   Google Scholar [25] 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 [26] H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, ().   Google Scholar [27] 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 [28] 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 [29] H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc. (3), 102 (2011), 50.  doi: 10.1112/plms/pdq013.  Google Scholar [30] 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 [31] 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 [32] H. Sumi and M. Urbański, Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups,, Discrete and Continuous Dynamical Systems Ser. A, 30 (2011), 313.  doi: 10.3934/dcds.2011.30.313.  Google Scholar [33] H. Sumi and M. Urbański, Transversality family of expanding rational semigroups,, preprint, (2011).   Google Scholar [34] A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps,, Invent. Math., 99 (1990), 627.  doi: 10.1007/BF01234434.  Google Scholar [35] W. Zhou 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] 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 [2] 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,, {\bf 19} (1999), 19 (1999), 1221.  doi: 10.1017/S0143385799141658.  Google Scholar [3] 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 [4] M. Büger, On the composition of polynomials of the form $z^2+c_n$,, Math. Ann., 310 (1998), 661.  doi: 10.1007/s002080050165.  Google Scholar [5] J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynam. Systems, 11 (1991), 687.  doi: 10.1017/S0143385700006428.  Google Scholar [6] 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 [7] 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 [8] M. Jonsson, Dynamics of polynomial skew products on $\C ^{2}$,, Math. Ann., 314 (1999), 403.  doi: 10.1007/s002080050301.  Google Scholar [9] M. Jonsson, Ergodic properties of fibered rational maps,, Ark. Mat., 38 (2000), 281.  doi: 10.1007/BF02384321.  Google Scholar [10] R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,'', Cambridge Tracts in Mathematics, 148 (2003).   Google Scholar [11] J. Milnor, "Dynamics in One Complex Variable," Third Edition,, Annals of Mathematical Studies, 160 (2006).   Google Scholar [12] F. Przytycki and M. Urbański, "Fractals in the Plane-The Ergodic Theory Methods,'', to be published from Cambridge University Press. Available from: \url{http://www.math.unt.edu/~urbanski/}., ().   Google Scholar [13] T. Ransford, "Potential Theory in the Complex Plane,", London Mathematical Society Student Texts, 28 (1995).   Google Scholar [14] O. Sester, Combinatorial configurations of fibered polynomials,, Ergodic Theory Dynam. Systems, 21 (2001), 915.   Google Scholar [15] R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357.   Google Scholar [16] R. Stankewitz and H. Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups,, Trans. Amer. Math. Soc., 363 (2011), 5293.  doi: 10.1090/S0002-9947-2011-05199-8.  Google Scholar [17] H. Sumi, On dynamics of hyperbolic rational semigroups,, J. Math. Kyoto Univ., 37 (1997), 717.   Google Scholar [18] 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 [19] 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 [20] H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563.   Google Scholar [21] H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390.  doi: 10.2996/kmj/1123767019.  Google Scholar [22] H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893.  doi: 10.1017/S0143385705000532.  Google Scholar [23] 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 [24] H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62.   Google Scholar [25] 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 [26] H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, ().   Google Scholar [27] 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 [28] 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 [29] H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc. (3), 102 (2011), 50.  doi: 10.1112/plms/pdq013.  Google Scholar [30] 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 [31] 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 [32] H. Sumi and M. Urbański, Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups,, Discrete and Continuous Dynamical Systems Ser. A, 30 (2011), 313.  doi: 10.3934/dcds.2011.30.313.  Google Scholar [33] H. Sumi and M. Urbański, Transversality family of expanding rational semigroups,, preprint, (2011).   Google Scholar [34] A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps,, Invent. Math., 99 (1990), 627.  doi: 10.1007/BF01234434.  Google Scholar [35] W. Zhou 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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