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 and Continuous Dynamical Systems, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313
References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs Vol. 50, American Mathematical Society, 1997.

[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), 347-372. doi: 10.2140/pjm.2001.198.347.

[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), 1221-1231. doi: 10.1017/S0143385799141658.

[4]

M. Büger, Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dynam. Systems, 17 (1997), 1289-1297. doi: 10.1017/S0143385797086458.

[5]

M. Büger, On the composition of polynomials of the form $z^2+c_n$, Math. Ann., 310 (1998), 661-683.

[6]

L. Carleson, P. W. Jones and J. -C. Yoccoz, Julia and John, Bol. Soc. Brazil. Math., 25 (1994), 1-30.

[7]

M. Denker and M. Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc., 328 (1991), 563-587. doi: 10.2307/2001795.

[8]

M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity, 4 (1991), 103-134. doi: 10.1088/0951-7715/4/1/008.

[9]

M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity, 4 (1991), 365-384. doi: 10.1088/0951-7715/4/2/008.

[10]

R. Devaney, "An Introduction to Chaotic Dynamical Systems," Reprint of the second (1989) edition. Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.

[11]

K. Falconer, "Techniques in Fractal Geometry," John Wiley & Sons, 1997.

[12]

H. Federer, "Geometric Measure Theory," Springer, 1969.

[13]

J. E. Fornaess and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708. doi: 10.1017/S0143385700006428.

[14]

Z. Gong, W. Qiu and Y. Li, Connectedness of Julia sets for a quadratic random dynamical system, Ergodic Theory Dynam. Systems, 23 (2003), 1807-1815. doi: 10.1017/S0143385703000129.

[15]

A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I, Proc. London Math. Soc. (3), 73 (1996), 358-384. doi: 10.1112/plms/s3-73.2.358.

[16]

A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups, Math. Z., 222 (1996), 161-169.

[17]

M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3 (1983), 351-386.

[18]

M. Martens, The existence of σ-finite invariant measures, Applications to real one-dimensional dynamics,, Front for the Math., (). 

[19]

P. Mattila, "Geometry of Sets and Measures in Euclidean spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.

[20]

R. D. Mauldin, T. Szarek and M. Urbański, Graph directed Markov systems on Hilbert spaces, Math. Proc. Cambridge Phil. Soc., 147 (2009), 455-488. doi: 10.1017/S0305004109002448.

[21]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105-154. doi: 10.1112/plms/s3-73.1.105.

[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.

[23]

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

[24]

V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry, preprint 2008, http://www.math.unt.edu/ urbanski/papers.html.

[25]

W. Parry, "Entropy and Generators in Ergodic Theory," Mathematics Lecture Note Series, 1969, Benjamin Inc., 1969.

[26]

F. Przytycki and M. Urbański, "Fractals in the Plane - The Ergodic Theory Methods,", to be published from Cambridge University Press, (). 

[27]

D. Ruelle, "Thermodynamic Formalism," Encyclopedia of Math. and Appl., 5, Addison-Wesley, Reading Mass., 1978.

[28]

R. Stankewitz, Completely invariant Julia sets of polynomial semigroups, Proc. Amer. Math. Soc., 127 (1999), 2889-2898. doi: 10.1090/S0002-9939-99-04857-1.

[29]

R. Stankewitz, Completely invariant sets of normality for rational semigroups, Complex Variables Theory Appl., 40 (2000), 199-210.

[30]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575. doi: 10.1090/S0002-9939-00-05313-2.

[31]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357-366.

[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., (). 

[33]

D. Steinsaltz, Random logistic maps and Lyapunov exponents, Indag. Mathem., N. S., 12 (2001), 557-584.

[34]

H. Sumi, On dynamics of hyperbolic rational semigroups, J. Math. Kyoto Univ., 37 (1997), 717-733.

[35]

H. Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups, Kodai Mathematical Journal, 21 (1998), 10-28. doi: 10.2996/kmj/1138043831.

[36]

H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019. doi: 10.1088/0951-7715/13/4/302.

[37]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.

[38]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Mathematical Journal, 28 (2005), 390-422; Also available from http://arxiv.org/abs/math/0405522.

[39]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922. doi: 10.1017/S0143385705000532.

[40]

H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane, Appl. Math. Comput., 187 (2007), 489-500. doi: 10.1016/j.amc.2006.08.149.

[41]

H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity, RIMS Kokyuroku, 1494 (2006), 62-86.

[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), 1205-1244. doi: 10.3934/dcds.2011.29.1205.

[43]

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

[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), 1869-1902. doi: 10.1017/S0143385709000923.

[45]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups, preprint 2007, http://arxiv.org/abs/math/0703591.

[46]

H. Sumi, Interaction cohomology of forward or backward self-similar systems, Adv. Math., 222 (2009), 729-781. doi: 10.1016/j.aim.2009.04.007.

[47]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. London Math. Soc., 102 (2011), 50-112. doi: 10.1112/plms/pdq013.

[48]

H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, preprint 2010, http://arxiv.org/abs/1008.3995.

[49]

H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps, Monatsh. Math., 156 (2009), 371-390. doi: 10.1007/s00605-008-0016-8.

[50]

H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups, Ergodic Theory Dynam. Systems, 30 (2010), 601-633. doi: 10.1017/S0143385709000297.

[51]

M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems, 14 (1994), 391-414.

[52]

M. Urbański, Geometry and ergodic theory of conformal non-recurrent dynamics, Ergodic Theory Dynam. Systems, 17 (1997), 1449-1476. doi: 10.1017/S014338579708646X.

[53]

P. Walters, "An Introduction to Ergodic Theory," Springer-Verlag, 1982.

[54]

W. Zhou and and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Sci. Bulletin, 37 (1992), 969-971.

show all references

References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs Vol. 50, American Mathematical Society, 1997.

[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), 347-372. doi: 10.2140/pjm.2001.198.347.

[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), 1221-1231. doi: 10.1017/S0143385799141658.

[4]

M. Büger, Self-similarity of Julia sets of the composition of polynomials, Ergodic Theory Dynam. Systems, 17 (1997), 1289-1297. doi: 10.1017/S0143385797086458.

[5]

M. Büger, On the composition of polynomials of the form $z^2+c_n$, Math. Ann., 310 (1998), 661-683.

[6]

L. Carleson, P. W. Jones and J. -C. Yoccoz, Julia and John, Bol. Soc. Brazil. Math., 25 (1994), 1-30.

[7]

M. Denker and M. Urbański, On the existence of conformal measures, Trans. Amer. Math. Soc., 328 (1991), 563-587. doi: 10.2307/2001795.

[8]

M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity, 4 (1991), 103-134. doi: 10.1088/0951-7715/4/1/008.

[9]

M. Denker and M. Urbański, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity, 4 (1991), 365-384. doi: 10.1088/0951-7715/4/2/008.

[10]

R. Devaney, "An Introduction to Chaotic Dynamical Systems," Reprint of the second (1989) edition. Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.

[11]

K. Falconer, "Techniques in Fractal Geometry," John Wiley & Sons, 1997.

[12]

H. Federer, "Geometric Measure Theory," Springer, 1969.

[13]

J. E. Fornaess and N. Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11 (1991), 687-708. doi: 10.1017/S0143385700006428.

[14]

Z. Gong, W. Qiu and Y. Li, Connectedness of Julia sets for a quadratic random dynamical system, Ergodic Theory Dynam. Systems, 23 (2003), 1807-1815. doi: 10.1017/S0143385703000129.

[15]

A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions I, Proc. London Math. Soc. (3), 73 (1996), 358-384. doi: 10.1112/plms/s3-73.2.358.

[16]

A. Hinkkanen and G. J. Martin, Julia Sets of Rational Semigroups, Math. Z., 222 (1996), 161-169.

[17]

M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems, 3 (1983), 351-386.

[18]

M. Martens, The existence of σ-finite invariant measures, Applications to real one-dimensional dynamics,, Front for the Math., (). 

[19]

P. Mattila, "Geometry of Sets and Measures in Euclidean spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.

[20]

R. D. Mauldin, T. Szarek and M. Urbański, Graph directed Markov systems on Hilbert spaces, Math. Proc. Cambridge Phil. Soc., 147 (2009), 455-488. doi: 10.1017/S0305004109002448.

[21]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105-154. doi: 10.1112/plms/s3-73.1.105.

[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.

[23]

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

[24]

V. Mayer, B. Skorulski and M. Urbański, Random distance expanding mappings, thermodynamic formalism, Gibbs measures, and fractal geometry, preprint 2008, http://www.math.unt.edu/ urbanski/papers.html.

[25]

W. Parry, "Entropy and Generators in Ergodic Theory," Mathematics Lecture Note Series, 1969, Benjamin Inc., 1969.

[26]

F. Przytycki and M. Urbański, "Fractals in the Plane - The Ergodic Theory Methods,", to be published from Cambridge University Press, (). 

[27]

D. Ruelle, "Thermodynamic Formalism," Encyclopedia of Math. and Appl., 5, Addison-Wesley, Reading Mass., 1978.

[28]

R. Stankewitz, Completely invariant Julia sets of polynomial semigroups, Proc. Amer. Math. Soc., 127 (1999), 2889-2898. doi: 10.1090/S0002-9939-99-04857-1.

[29]

R. Stankewitz, Completely invariant sets of normality for rational semigroups, Complex Variables Theory Appl., 40 (2000), 199-210.

[30]

R. Stankewitz, Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's, Proc. Amer. Math. Soc., 128 (2000), 2569-2575. doi: 10.1090/S0002-9939-00-05313-2.

[31]

R. Stankewitz, T. Sugawa and H. Sumi, Some counterexamples in dynamics of rational semigroups, Annales Academiae Scientiarum Fennicae Mathematica, 29 (2004), 357-366.

[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., (). 

[33]

D. Steinsaltz, Random logistic maps and Lyapunov exponents, Indag. Mathem., N. S., 12 (2001), 557-584.

[34]

H. Sumi, On dynamics of hyperbolic rational semigroups, J. Math. Kyoto Univ., 37 (1997), 717-733.

[35]

H. Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups, Kodai Mathematical Journal, 21 (1998), 10-28. doi: 10.2996/kmj/1138043831.

[36]

H. Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13 (2000), 995-1019. doi: 10.1088/0951-7715/13/4/302.

[37]

H. Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21 (2001), 563-603.

[38]

H. Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Mathematical Journal, 28 (2005), 390-422; Also available from http://arxiv.org/abs/math/0405522.

[39]

H. Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26 (2006), 893-922. doi: 10.1017/S0143385705000532.

[40]

H. Sumi, Random dynamics of polynomials and devil's-staircase-like functions in the complex plane, Appl. Math. Comput., 187 (2007), 489-500. doi: 10.1016/j.amc.2006.08.149.

[41]

H. Sumi, The space of postcritically bounded 2-generator polynomial semigroups with hyperbolicity, RIMS Kokyuroku, 1494 (2006), 62-86.

[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), 1205-1244. doi: 10.3934/dcds.2011.29.1205.

[43]

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

[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), 1869-1902. doi: 10.1017/S0143385709000923.

[45]

H. Sumi, Dynamics of postcritically bounded polynomial semigroups, preprint 2007, http://arxiv.org/abs/math/0703591.

[46]

H. Sumi, Interaction cohomology of forward or backward self-similar systems, Adv. Math., 222 (2009), 729-781. doi: 10.1016/j.aim.2009.04.007.

[47]

H. Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. London Math. Soc., 102 (2011), 50-112. doi: 10.1112/plms/pdq013.

[48]

H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, preprint 2010, http://arxiv.org/abs/1008.3995.

[49]

H. Sumi and M. Urbański, The equilibrium states for semigroups of rational maps, Monatsh. Math., 156 (2009), 371-390. doi: 10.1007/s00605-008-0016-8.

[50]

H. Sumi and M. Urbański, Real analyticity of Hausdorff dimension for expanding rational semigroups, Ergodic Theory Dynam. Systems, 30 (2010), 601-633. doi: 10.1017/S0143385709000297.

[51]

M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems, 14 (1994), 391-414.

[52]

M. Urbański, Geometry and ergodic theory of conformal non-recurrent dynamics, Ergodic Theory Dynam. Systems, 17 (1997), 1449-1476. doi: 10.1017/S014338579708646X.

[53]

P. Walters, "An Introduction to Ergodic Theory," Springer-Verlag, 1982.

[54]

W. Zhou and and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Sci. Bulletin, 37 (1992), 969-971.

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