Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method

Jaerisch, Johannes; Sumi, Hiroki

doi:10.1090/tran/6862

Dynamics of infinitely generated nicely expanding rational semigroups and the inducing method
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Trans. Amer. Math. Soc. 369 (2017), 6147-6187 Request permission

Abstract:

We investigate the dynamics of semigroups of rational maps on the Riemann sphere. To establish a fractal theory of the Julia sets of infinitely generated semigroups of rational maps, we introduce a new class of semigroups which we call nicely expanding rational semigroups. More precisely, we prove Bowen’s formula for the Hausdorff dimension of the pre-Julia sets, which we also introduce in this paper. We apply our results to the study of the Julia sets of non-hyperbolic rational semigroups. For these results, we do not assume the cone condition, which has been assumed in the study of infinite contracting iterated function systems. Similarly, we show that Bowen’s formula holds for the limit set of a contracting conformal iterated function system without the cone condition.

References

Jon Aaronson, Manfred Denker, and Mariusz Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps, Trans. Amer. Math. Soc. 337 (1993), no. 2, 495–548. MR 1107025, DOI 10.1090/S0002-9947-1993-1107025-2
Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
Alan F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics, vol. 132, Springer-Verlag, New York, 1991. Complex analytic dynamical systems. MR 1128089, DOI 10.1007/978-1-4612-4422-6
Rufus Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 11–25. MR 556580, DOI 10.1007/BF02684767
Rainer Brück, Matthias Büger, and Stefan Reitz, Random iterations of polynomials of the form $z^2+c_n$: connectedness of Julia sets, Ergodic Theory Dynam. Systems 19 (1999), no. 5, 1221–1231. MR 1721617, DOI 10.1017/S0143385799141658
Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
John Erik Fornæss and Nessim Sibony, Random iterations of rational functions, Ergodic Theory Dynam. Systems 11 (1991), no. 4, 687–708. MR 1145616, DOI 10.1017/S0143385700006428
Jacek Graczyk and Stanislav Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math. 175 (2009), no. 2, 335–415. MR 2470110, DOI 10.1007/s00222-008-0152-8
B. M. Gurevič, Topological entropy of a countable Markov chain, Dokl. Akad. Nauk SSSR 187 (1969), 715–718 (Russian). MR 0263162
A. Hinkkanen and G. J. Martin, The dynamics of semigroups of rational functions. I, Proc. London Math. Soc. (3) 73 (1996), no. 2, 358–384. MR 1397693, DOI 10.1112/plms/s3-73.2.358
Johannes Jaerisch, Thermodynamic formalism for group-extended Markov systems with applications to Fuchsian groups, Doctoral Dissertation at the University Bremen (2011).
Johannes Jaerisch, Marc Kesseböhmer, and Sanaz Lamei, Induced topological pressure for countable state Markov shifts, Stoch. Dyn. 14 (2014), no. 2, 1350016, 31. MR 3190211, DOI 10.1142/S0219493713500160
John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
R. Daniel Mauldin and Mariusz Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3) 73 (1996), no. 1, 105–154. MR 1387085, DOI 10.1112/plms/s3-73.1.105
R. D. Mauldin and M. Urbański, Parabolic iterated function systems, Ergodic Theory Dynam. Systems 20 (2000), no. 5, 1423–1447. MR 1786722, DOI 10.1017/S0143385700000778
R. D. Mauldin and M. Urbański, Graph directed Markov systems, Cambridge Tracts in Mathematics, vol. 148, Cambridge, 2003.
Volker Mayer, Bartlomiej Skorulski, and Mariusz Urbanski, Distance expanding random mappings, thermodynamical formalism, Gibbs measures and fractal geometry, Lecture Notes in Mathematics, vol. 2036, Springer, Heidelberg, 2011. MR 2866474, DOI 10.1007/978-3-642-23650-1
Feliks Przytycki, Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: on non-renormalizable quadratic polynomials, Trans. Amer. Math. Soc. 350 (1998), no. 2, 717–742. MR 1407501, DOI 10.1090/S0002-9947-98-01890-X
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683, DOI 10.1515/9781400873173
David Ruelle, Statistical mechanics on a compact set with $Z^{v}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc. 187 (1973), 237–251. MR 417391, DOI 10.1090/S0002-9947-1973-0417391-6
David Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 99–107. MR 684247, DOI 10.1017/s0143385700009603
Omri M. Sarig, Thermodynamic formalism for countable Markov shifts, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1565–1593. MR 1738951, DOI 10.1017/S0143385799146820
F. Schweiger, Numbertheoretical endomorphisms with $\sigma$-finite invariant measure, Israel J. Math. 21 (1975), no. 4, 308–318. MR 384735, DOI 10.1007/BF02757992
Rich Stankewitz, Density of repelling fixed points in the Julia set of a rational or entire semigroup, II, Discrete Contin. Dyn. Syst. 32 (2012), no. 7, 2583–2589. MR 2900562, DOI 10.3934/dcds.2012.32.2583
Rich Stankewitz and Hiroki Sumi, Dynamical properties and structure of Julia sets of postcritically bounded polynomial semigroups, Trans. Amer. Math. Soc. 363 (2011), no. 10, 5293–5319. MR 2813416, DOI 10.1090/S0002-9947-2011-05199-8
Hiroki Sumi, On dynamics of hyperbolic rational semigroups, J. Math. Kyoto Univ. 37 (1997), no. 4, 717–733. MR 1625944, DOI 10.1215/kjm/1250518211
Hiroki Sumi, On Hausdorff dimension of Julia sets of hyperbolic rational semigroups, Kodai Math. J. 21 (1998), no. 1, 10–28. MR 1625124, DOI 10.2996/kmj/1138043831
Hiroki Sumi, Skew product maps related to finitely generated rational semigroups, Nonlinearity 13 (2000), no. 4, 995–1019. MR 1767945, DOI 10.1088/0951-7715/13/4/302
Hiroki Sumi, Dynamics of sub-hyperbolic and semi-hyperbolic rational semigroups and skew products, Ergodic Theory Dynam. Systems 21 (2001), no. 2, 563–603. MR 1827119, DOI 10.1017/S0143385701001286
Hiroki Sumi, Dimensions of Julia sets of expanding rational semigroups, Kodai Math. J. 28 (2005), no. 2, 390–422. MR 2153926, DOI 10.2996/kmj/1123767019
Hiroki Sumi, Semi-hyperbolic fibered rational maps and rational semigroups, Ergodic Theory Dynam. Systems 26 (2006), no. 3, 893–922. MR 2237476, DOI 10.1017/S0143385705000532
Hiroki Sumi, Interaction cohomology of forward or backward self-similar systems, Adv. Math. 222 (2009), no. 3, 729–781. MR 2553369, DOI 10.1016/j.aim.2009.04.007
Hiroki 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), no. 6, 1869–1902. MR 2736899, DOI 10.1017/S0143385709000923
Hiroki Sumi, Rational semigroups, random complex dynamics and singular functions on the complex plane, Amer. Math. Soc. Transl. 230 (2010), no. 2, 161–200.
Hiroki Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. Lond. Math. Soc. (3) 102 (2011), no. 1, 50–112. MR 2747724, DOI 10.1112/plms/pdq013
Hiroki Sumi, Dynamics of postcritically bounded polynomial semigroups I: connected components of the Julia sets, Discrete Contin. Dyn. Syst. 29 (2011), no. 3, 1205–1244. MR 2773173, DOI 10.3934/dcds.2011.29.1205
Hiroki Sumi, Random complex dynamics and devil’s coliseums, Nonlinearity 28 (2015), no. 4, 1135–1161. MR 3333714, DOI 10.1088/0951-7715/28/4/1135
Hiroki Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, Adv. Math. 245 (2013), 137–181. MR 3084426, DOI 10.1016/j.aim.2013.05.023
Hiroki Sumi, The space of 2-generator postcritically bounded polynomial semigroups and random complex dynamics, Adv. Math. 290 (2016), 809–859. MR 3451939, DOI 10.1016/j.aim.2015.12.011
Hiroki Sumi and Mariusz Urbański, Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups, Discrete Contin. Dyn. Syst. 30 (2011), no. 1, 313–363. MR 2773145, DOI 10.3934/dcds.2011.30.313
Hiroki Sumi and Mariusz Urbański, Transversality family of expanding rational semigroups, Adv. Math. 234 (2013), 697–734. MR 3003942, DOI 10.1016/j.aim.2012.10.020
Mariusz Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 391–414. MR 1279476, DOI 10.1017/S0143385700007926
Peter Walters, Convergence of the Ruelle operator for a function satisfying Bowen’s condition, Trans. Amer. Math. Soc. 353 (2001), no. 1, 327–347. MR 1783787, DOI 10.1090/S0002-9947-00-02656-8
Peter Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1975), no. 4, 937–971. MR 390180, DOI 10.2307/2373682
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108, DOI 10.1007/978-1-4612-5775-2
W. Zhou and F. Ren, The Julia sets of the random iteration of rational functions, Chinese Sci. Bull., 37 (12), 1992, p969-971.