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:
[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.

[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.

[3]

M. Büger, Self-similarity of Julia sets of the composition of polynomials,, Ergodic Theory Dynam. Systems, 17 (1997), 1289. doi: 10.1017/S0143385797086458.

[4]

M. Büger, On the composition of polynomials of the form $z^2+c_n$,, Math. Ann., 310 (1998), 661. doi: 10.1007/s002080050165.

[5]

J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynam. Systems, 11 (1991), 687. doi: 10.1017/S0143385700006428.

[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.

[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.

[8]

M. Jonsson, Dynamics of polynomial skew products on $\C ^{2}$,, Math. Ann., 314 (1999), 403. doi: 10.1007/s002080050301.

[9]

M. Jonsson, Ergodic properties of fibered rational maps,, Ark. Mat., 38 (2000), 281. doi: 10.1007/BF02384321.

[10]

R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,'', Cambridge Tracts in Mathematics, 148 (2003).

[11]

J. Milnor, "Dynamics in One Complex Variable," Third Edition,, Annals of Mathematical Studies, 160 (2006).

[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/}., ().

[13]

T. Ransford, "Potential Theory in the Complex Plane,", London Mathematical Society Student Texts, 28 (1995).

[14]

O. Sester, Combinatorial configurations of fibered polynomials,, Ergodic Theory Dynam. Systems, 21 (2001), 915.

[15]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357.

[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.

[17]

H. Sumi, On dynamics of hyperbolic rational semigroups,, J. Math. Kyoto Univ., 37 (1997), 717.

[18]

H. Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups,, Kodai Mathematical Journal, 21 (1998), 10. doi: 10.2996/kmj/1138043831.

[19]

H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995. doi: 10.1088/0951-7715/13/4/302.

[20]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563.

[21]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390. doi: 10.2996/kmj/1123767019.

[22]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893. doi: 10.1017/S0143385705000532.

[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.

[24]

H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62.

[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.

[26]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, ().

[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.

[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.

[29]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc. (3), 102 (2011), 50. doi: 10.1112/plms/pdq013.

[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.

[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.

[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.

[33]

H. Sumi and M. Urbański, Transversality family of expanding rational semigroups,, preprint, (2011).

[34]

A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps,, Invent. Math., 99 (1990), 627. doi: 10.1007/BF01234434.

[35]

W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions,, Chinese Sci. Bulletin., 37 (1992), 969.

show all references

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.

[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.

[3]

M. Büger, Self-similarity of Julia sets of the composition of polynomials,, Ergodic Theory Dynam. Systems, 17 (1997), 1289. doi: 10.1017/S0143385797086458.

[4]

M. Büger, On the composition of polynomials of the form $z^2+c_n$,, Math. Ann., 310 (1998), 661. doi: 10.1007/s002080050165.

[5]

J. E. Fornaess and N. Sibony, Random iterations of rational functions,, Ergodic Theory Dynam. Systems, 11 (1991), 687. doi: 10.1017/S0143385700006428.

[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.

[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.

[8]

M. Jonsson, Dynamics of polynomial skew products on $\C ^{2}$,, Math. Ann., 314 (1999), 403. doi: 10.1007/s002080050301.

[9]

M. Jonsson, Ergodic properties of fibered rational maps,, Ark. Mat., 38 (2000), 281. doi: 10.1007/BF02384321.

[10]

R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,'', Cambridge Tracts in Mathematics, 148 (2003).

[11]

J. Milnor, "Dynamics in One Complex Variable," Third Edition,, Annals of Mathematical Studies, 160 (2006).

[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/}., ().

[13]

T. Ransford, "Potential Theory in the Complex Plane,", London Mathematical Society Student Texts, 28 (1995).

[14]

O. Sester, Combinatorial configurations of fibered polynomials,, Ergodic Theory Dynam. Systems, 21 (2001), 915.

[15]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups,, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357.

[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.

[17]

H. Sumi, On dynamics of hyperbolic rational semigroups,, J. Math. Kyoto Univ., 37 (1997), 717.

[18]

H. Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups,, Kodai Mathematical Journal, 21 (1998), 10. doi: 10.2996/kmj/1138043831.

[19]

H. Sumi, Skew product maps related to finitely generated rational semigroups,, Nonlinearity, 13 (2000), 995. doi: 10.1088/0951-7715/13/4/302.

[20]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products,, Ergodic Theory Dynam. Systems, 21 (2001), 563.

[21]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups,, Kodai Mathematical Journal, 28 (2005), 390. doi: 10.2996/kmj/1123767019.

[22]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups,, Ergodic Theory Dynam. Systems, 26 (2006), 893. doi: 10.1017/S0143385705000532.

[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.

[24]

H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity,, RIMS Kokyuroku, 1494 (2006), 62.

[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.

[26]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups II: Fiberwise dynamics and the Julia sets,, preprint, ().

[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.

[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.

[29]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps,, Proc. London Math. Soc. (3), 102 (2011), 50. doi: 10.1112/plms/pdq013.

[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.

[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.

[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.

[33]

H. Sumi and M. Urbański, Transversality family of expanding rational semigroups,, preprint, (2011).

[34]

A. Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps,, Invent. Math., 99 (1990), 627. doi: 10.1007/BF01234434.

[35]

W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions,, Chinese Sci. Bulletin., 37 (1992), 969.

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