Almost sure invariance principle for non-autonomous holomorphic dynamics in ℙ^{𝕜}

Bayraktar, Turgay

doi:10.1090/ecgd/319

Almost sure invariance principle for non-autonomous holomorphic dynamics in $\mathbb {P}^k$
HTML articles powered by AMS MathViewer

by Turgay Bayraktar

Conform. Geom. Dyn. 22 (2018), 45-61

DOI: https://doi.org/10.1090/ecgd/319

Published electronically: May 31, 2018

PDF | Request permission

Abstract:

We prove an almost sure invariance principle, a strong form of approximation by Brownian motion, for non-autonomous holomorphic dynamical systems on complex projective space $\mathbb {P}^k$ for Hölder continuous and DSH observables.

References

Turgay Bayraktar, Ergodic properties of random holomorphic endomorphisms of $\Bbb P^k$, Int. Math. Res. Not. IMRN 4 (2015), 927–959. MR 3340342, DOI 10.1093/imrn/rnt226
Turgay Bayraktar, Corrigendum to: Ergodic properties of random holomorphic endomorphisms of $\Bbb {P}^k$ [ MR3340342], Int. Math. Res. Not. IMRN 14 (2015), 6005–6009. MR 3384466, DOI 10.1093/imrn/rnv168
Jean-Yves Briend and Julien Duval, Deux caractérisations de la mesure d’équilibre d’un endomorphisme de $\textrm {P}^k(\mathbf C)$, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 145–159 (French, with English and French summaries). MR 1863737, DOI 10.1007/s10240-001-8190-4
Hans Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103–144 (1965). MR 194595, DOI 10.1007/BF02591353
Serge Cantat and Stéphane Le Borgne, Théorème limite central pour les endomorphismes holomorphes et les correspondances modulaires, Int. Math. Res. Not. 56 (2005), 3479–3510 (French). MR 2200586, DOI 10.1155/IMRN.2005.3479
Christophe Cuny and Florence Merlevède, Strong invariance principles with rate for “reverse” martingale differences and applications, J. Theoret. Probab. 28 (2015), no. 1, 137–183. MR 3320963, DOI 10.1007/s10959-013-0506-z
Jean-Pierre Conze and Albert Raugi, Limit theorems for sequential expanding dynamical systems on $[0,1]$, Ergodic theory and related fields, Contemp. Math., vol. 430, Amer. Math. Soc., Providence, RI, 2007, pp. 89–121. MR 2331327, DOI 10.1090/conm/430/08253
Tien-Cuong Dinh and Christophe Dupont, Dimension de la mesure d’équilibre d’applications méromorphes, J. Geom. Anal. 14 (2004), no. 4, 613–627 (French, with English summary). MR 2111420, DOI 10.1007/BF02922172
Tien-Cuong Dinh and Viêt-Anh Nguyên, Characterization of Monge-Ampère measures with Hölder continuous potentials, J. Funct. Anal. 266 (2014), no. 1, 67–84. MR 3121721, DOI 10.1016/j.jfa.2013.08.026
Tien-Cuong Dinh, Viêt-Anh Nguyên, and Nessim Sibony, Exponential estimates for plurisubharmonic functions and stochastic dynamics, J. Differential Geom. 84 (2010), no. 3, 465–488. MR 2669362
Tien-Cuong Dinh and Nessim Sibony, Decay of correlations and the central limit theorem for meromorphic maps, Comm. Pure Appl. Math. 59 (2006), no. 5, 754–768. MR 2172806, DOI 10.1002/cpa.20119
Tien-Cuong Dinh and Nessim Sibony, Distribution des valeurs de transformations méromorphes et applications, Comment. Math. Helv. 81 (2006), no. 1, 221–258 (French, with English summary). MR 2208805, DOI 10.4171/CMH/50
Henry de Thélin, Endomorphismes pseudo-aléatoires dans les espaces projectifs I, Manuscripta Math. 142 (2013), no. 3-4, 347–367 (French, with English and French summaries). MR 3117166, DOI 10.1007/s00229-012-0603-9
Christophe Dupont, Bernoulli coding map and almost sure invariance principle for endomorphisms of $\Bbb P^k$, Probab. Theory Related Fields 146 (2010), no. 3-4, 337–359. MR 2574731, DOI 10.1007/s00440-008-0192-4
Rick Durrett, Probability: theory and examples, 4th ed., Cambridge Series in Statistical and Probabilistic Mathematics, vol. 31, Cambridge University Press, Cambridge, 2010. MR 2722836, DOI 10.1017/CBO9780511779398
John Erik Fornaess and Nessim Sibony, Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992) Ann. of Math. Stud., vol. 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 135–182. MR 1369137
I. M. Gel′fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1264417, DOI 10.1007/978-0-8176-4771-1
M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739–741 (Russian). MR 0251785
Nicolai Haydn, Convergence of the transfer operator for rational maps, Ergodic Theory Dynam. Systems 19 (1999), no. 3, 657–669. MR 1695914, DOI 10.1017/S0143385799130190
Nicolai Haydn, Matthew Nicol, Andrew Török, and Sandro Vaienti, Almost sure invariance principle for sequential and non-stationary dynamical systems, Trans. Amer. Math. Soc. 369 (2017), no. 8, 5293–5316. MR 3646763, DOI 10.1090/tran/6812
John H. Hubbard and Peter Papadopol, Superattractive fixed points in $\textbf {C}^n$, Indiana Univ. Math. J. 43 (1994), no. 1, 321–365. MR 1275463, DOI 10.1512/iumj.1994.43.43014
M. Ju. Ljubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), no. 3, 351–385. MR 741393, DOI 10.1017/S0143385700002030
A. Korepanov, Z. Kosloff, and I. Melbourne, Martingale-coboundary decomposition for families of dynamical systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire 35 (2018), no. 4, 859–885. MR 3795019, DOI 10.1016/j.anihpc.2017.08.005
Ian Melbourne and Matthew Nicol, Almost sure invariance principle for nonuniformly hyperbolic systems, Comm. Math. Phys. 260 (2005), no. 1, 131–146. MR 2175992, DOI 10.1007/s00220-005-1407-5
Han Peters, Non-autonomous dynamics in $\Bbb P^k$, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1295–1304. MR 2158406, DOI 10.1017/S0143385704000987
Feliks Przytycki and Juan Rivera-Letelier, Statistical properties of topological Collet-Eckmann maps, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 1, 135–178 (English, with English and French summaries). MR 2332354, DOI 10.1016/j.ansens.2006.11.002
Walter Philipp and William Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. 2 (1975), no. 161,, 161, iv+140. MR 433597, DOI 10.1090/memo/0161
Feliks Przytycki, Mariusz Urbański, and Anna Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I, Ann. of Math. (2) 130 (1989), no. 1, 1–40. MR 1005606, DOI 10.2307/1971475
Nessim Sibony, Dynamique des applications rationnelles de $\mathbf P^k$, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix–x, xi–xii, 97–185 (French, with English and French summaries). MR 1760844
Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503903