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Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method. (English) Zbl 1373.30034

Let Rat be the set of all non-constant rational maps on the Riemann sphere \(\hat{\mathbb{C}}\). A subsemigroup of Rat with semigroup operation being the functional composition is called a rational semigroup. Two new useful notions are defined in this long study: nicely expanding rational semigroup and pre-Fatou and pre-Julia sets. The main results are Theorem 6.5 and Proposition 6.3 which establish a Bowen-type formula for pre-Julia sets, where the case of equality is provided by the open set condition.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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[1] Aaronson, Jon; Denker, Manfred; Urba{\'n}ski, Mariusz, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc., 337, 2, 495-548 (1993) · Zbl 0789.28010 · doi:10.2307/2154231
[2] Ahlfors, Lars V., Complex analysis, xi+331 pp. (1978), McGraw-Hill Book Co., New York
[3] Beardon, Alan F., Iteration of rational functions, Graduate Texts in Mathematics 132, xvi+280 pp. (1991), Springer-Verlag, New York · Zbl 0742.30002 · doi:10.1007/978-1-4612-4422-6
[4] Bowen, Rufus, Hausdorff dimension of quasicircles, Inst. Hautes \'Etudes Sci. Publ. Math., 50, 11-25 (1979) · Zbl 0439.30032
[5] Br{\`“u}ck, Rainer; B{\'”u}ger, Matthias; Reitz, Stefan, Random iterations of polynomials of the form \(z^2+c_n\): connectedness of Julia sets, Ergodic Theory Dynam. Systems, 19, 5, 1221-1231 (1999) · Zbl 0942.37041 · doi:10.1017/S0143385799141658
[6] Falconer, Kenneth, Fractal geometry, xxviii+337 pp. (2003), John Wiley & Sons, Inc., Hoboken, NJ · Zbl 1060.28005 · doi:10.1002/0470013850
[7] Federer, Herbert, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, xiv+676 pp. (1969), Springer-Verlag New York Inc., New York · Zbl 0176.00801
[8] Forn{\ae }ss, John Erik; Sibony, Nessim, Random iterations of rational functions, Ergodic Theory Dynam. Systems, 11, 4, 687-708 (1991) · Zbl 0753.30019 · doi:10.1017/S0143385700006428
[9] Graczyk, Jacek; Smirnov, Stanislav, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175, 2, 335-415 (2009) · Zbl 1163.37008 · doi:10.1007/s00222-008-0152-8
[10] Gurevi{\v{c}}, B. M., Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR, 187, 715-718 (1969) · Zbl 0194.49602
[11] Hinkkanen, Aimo; Martin, Gaven J., The dynamics of semigroups of rational functions. I, Proc. London Math. Soc. (3), 73, 2, 358-384 (1996) · Zbl 0859.30026 · doi:10.1112/plms/s3-73.2.358
[12] Johannes Jaerisch, Thermodynamic formalism for group-extended Markov systems with applications to Fuchsian groups, Doctoral Dissertation at the University Bremen (2011).
[13] Jaerisch, Johannes; Kesseb{\"o}hmer, Marc; Lamei, Sanaz, Induced topological pressure for countable state Markov shifts, Stoch. Dyn., 14, 2, 1350016, 31 pp. (2014) · Zbl 1376.37074 · doi:10.1142/S0219493713500160
[14] Milnor, John, Dynamics in one complex variable, Annals of Mathematics Studies 160, viii+304 pp. (2006), Princeton University Press, Princeton, NJ · Zbl 1085.30002
[15] Mauldin, R. Daniel; Urba{\'n}ski, Mariusz, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73, 1, 105-154 (1996) · Zbl 0852.28005 · doi:10.1112/plms/s3-73.1.105
[16] Mauldin, R. D.; Urba{\'n}ski, M., Parabolic iterated function systems, Ergodic Theory Dynam. Systems, 20, 5, 1423-1447 (2000) · Zbl 0982.37045 · doi:10.1017/S0143385700000778
[17] R. D. Mauldin and M. Urba\'nski, Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge, 2003. · Zbl 1033.37025
[18] Mayer, Volker; Skorulski, Bartlomiej; Urbanski, Mariusz, Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry, Lecture Notes in Mathematics 2036, x+112 pp. (2011), Springer, Heidelberg · Zbl 1237.37003 · doi:10.1007/978-3-642-23650-1
[19] Przytycki, Feliks, Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials, Trans. Amer. Math. Soc., 350, 2, 717-742 (1998) · Zbl 0892.58063 · doi:10.1090/S0002-9947-98-01890-X
[20] Rockafellar, R. Tyrrell, Convex analysis, Princeton Mathematical Series, No. 28, xviii+451 pp. (1970), Princeton University Press, Princeton, N.J. · Zbl 0932.90001
[21] Ruelle, David, Statistical mechanics on a compact set with \(Z^v\) action satisfying expansiveness and specification, Trans. Amer. Math. Soc., 187, 237-251 (1973) · Zbl 0278.28012
[22] Ruelle, David, Repellers for real analytic maps, Ergodic Theory Dynamical Systems, 2, 1, 99-107 (1982) · Zbl 0506.58024
[23] Sarig, Omri M., Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems, 19, 6, 1565-1593 (1999) · Zbl 0994.37005 · doi:10.1017/S0143385799146820
[24] Schweiger, F., Numbertheoretical endomorphisms with \(\sigma \)-finite invariant measure, Israel J. Math., 21, 4, 308-318 (1975) · Zbl 0314.10037
[25] Stankewitz, Rich, Density of repelling fixed points in the Julia set of a rational or entire semigroup, II, Discrete Contin. Dyn. Syst., 32, 7, 2583-2589 (2012) · Zbl 1267.37046 · doi:10.3934/dcds.2012.32.2583
[26] Stankewitz, Rich; Sumi, Hiroki, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups, Trans. Amer. Math. Soc., 363, 10, 5293-5319 (2011) · Zbl 1247.37042 · doi:10.1090/S0002-9947-2011-05199-8
[27] Sumi, Hiroki, On dynamics of hyperbolic rational semigroups, J. Math. Kyoto Univ., 37, 4, 717-733 (1997) · Zbl 0938.37026
[28] Sumi, Hiroki, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups, Kodai Math. J., 21, 1, 10-28 (1998) · Zbl 0917.58038 · doi:10.2996/kmj/1138043831
[29] Sumi, Hiroki, Skew product maps related to finitely generated rational semigroups, Nonlinearity, 13, 4, 995-1019 (2000) · Zbl 0959.30014 · doi:10.1088/0951-7715/13/4/302
[30] Sumi, Hiroki, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems, 21, 2, 563-603 (2001) · Zbl 0993.37022 · doi:10.1017/S0143385701001286
[31] Sumi, Hiroki, Dimensions of Julia sets of expanding rational semigroups, Kodai Math. J., 28, 2, 390-422 (2005) · Zbl 1092.37027 · doi:10.2996/kmj/1123767019
[32] Sumi, Hiroki, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems, 26, 3, 893-922 (2006) · Zbl 1095.37017 · doi:10.1017/S0143385705000532
[33] Sumi, Hiroki, Interaction cohomology of forward or backward self-similar systems, Adv. Math., 222, 3, 729-781 (2009) · Zbl 1180.37054 · doi:10.1016/j.aim.2009.04.007
[34] Sumi, Hiroki, 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, 6, 1869-1902 (2010) · Zbl 1219.37037 · doi:10.1017/S0143385709000923
[35] Hiroki Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane, Amer. Math. Soc. Transl. 230 (2010), no. 2, 161-200.
[36] Sumi, Hiroki, Random complex dynamics and semigroups of holomorphic maps, Proc. Lond. Math. Soc. (3), 102, 1, 50-112 (2011) · Zbl 1222.37041 · doi:10.1112/plms/pdq013
[37] Sumi, Hiroki, Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets, Discrete Contin. Dyn. Syst., 29, 3, 1205-1244 (2011) · Zbl 1216.37015 · doi:10.3934/dcds.2011.29.1205
[38] Sumi, Hiroki, Random complex dynamics and devil’s coliseums, Nonlinearity, 28, 4, 1135-1161 (2015) · Zbl 1334.37040 · doi:10.1088/0951-7715/28/4/1135
[39] Sumi, Hiroki, Cooperation principle, stability and bifurcation in random complex dynamics, Adv. Math., 245, 137-181 (2013) · Zbl 1329.37045 · doi:10.1016/j.aim.2013.05.023
[40] Sumi, Hiroki, The space of 2-generator postcritically bounded polynomial semigroups and random complex dynamics, Adv. Math., 290, 809-859 (2016) · Zbl 1408.37081 · doi:10.1016/j.aim.2015.12.011
[41] Sumi, Hiroki; Urba{\'n}ski, Mariusz, Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups, Discrete Contin. Dyn. Syst., 30, 1, 313-363 (2011) · Zbl 1218.37058 · doi:10.3934/dcds.2011.30.313
[42] Sumi, Hiroki; Urba{\'n}ski, Mariusz, Transversality family of expanding rational semigroups, Adv. Math., 234, 697-734 (2013) · Zbl 1334.37043 · doi:10.1016/j.aim.2012.10.020
[43] Urba{\'n}ski, Mariusz, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems, 14, 2, 391-414 (1994) · Zbl 0807.58025 · doi:10.1017/S0143385700007926
[44] Walters, Peter, Convergence of the Ruelle operator for a function satisfying Bowen’s condition, Trans. Amer. Math. Soc., 353, 1, 327-347 (electronic) (2001) · Zbl 1050.37015 · doi:10.1090/S0002-9947-00-02656-8
[45] Walters, Peter, A variational principle for the pressure of continuous transformations, Amer. J. Math., 97, 4, 937-971 (1975) · Zbl 0318.28007
[46] Walters, Peter, An introduction to ergodic theory, Graduate Texts in Mathematics 79, ix+250 pp. (1982), Springer-Verlag, New York-Berlin · Zbl 0958.28011
[47] W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Sci. Bull., 37 (12), 1992, p969-971. · Zbl 0768.58015
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