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Bernoulli coding map and almost sure invariance principle for endomorphisms of \({\mathbb P^k}\)

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Let f be an holomorphic endomorphism of \({\mathbb{P}^k}\) and μ be its measure of maximal entropy. We prove an almost sure invariance principle for the systems \({(\mathbb{P}^k,f,\mu)}\). Our class \({\mathcal {U}}\) of observables includes the Hölder functions and unbounded ones which present analytic singularities. The proof is based on a geometric construction of a Bernoulli coding map \({\omega: (\Sigma, s, \nu) \to (\mathbb{P}^k,f,\mu)}\). We obtain the invariance principle for an observable ψ on \({(\mathbb{P}^k,f,\mu)}\) by applying Philipp–Stout’s theorem for \({\chi = \psi \circ \omega}\) on (Σ, s, ν). The invariance principle implies the central limit theorem as well as several statistical properties for the class \({\mathcal {U}}\). As an application, we give a direct proof of the absolute continuity of the measure μ when it satisfies Pesin’s formula. This approach relies on the central limit theorem for the unbounded observable log \({{\tt Jac}\, f \in \mathcal{U}}\).

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  1. 1

    Ahlfors, L.: Lectures on quasiconformal mappings, 2nd edn. University Lecture Series, 38. American Mathematical Society, Providence (2006)

  2. 2

    Berteloot F., Dupont C.: Une caractérisation des endomorphismes de Lattès par leur mesure de Green. Comment. Math. Helv. 80(2), 433–454 (2005)

  3. 3

    Berteloot F., Loeb J.J.: Une caractérisation géométrique des exemples de Lattès de \(\mathbb {CP}^k\). Bull. Soc. Math. Fr. 129(2), 175–188 (2001)

  4. 4

    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol. 470 (1975)

  5. 5

    Briend J.Y.: La propriété de Bernoulli pour les endomorphismes de \({\mathbb {P}^k}(\mathbb {C})\). Ergodic Theory Dyn. Syst. 22(2), 323–327 (2002)

  6. 6

    Briend J.Y., Duval J.: Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de \(\mathbb {CP}^k\). Acta Math. 182(2), 143–157 (1999)

  7. 7

    Briend J.Y., Duval J.: Deux caractérisations de la mesure d’équilibre d’un endomorphisme de \(\mathbb {CP}^k\). Publ. Math. Inst. Hautes Études Sci. 93, 145–159 (2001)

  8. 8

    Buzzi J.: The coding of non-uniformly expanding maps with application to endomorphisms of \({\mathbb {CP}^k}\). Ergodic Theory Dyn. Syst. 23(4), 1015–1024 (2003)

  9. 9

    Cantat S., Leborgne S.: Théorème limite central pour les endomorphismes holomorphes et les correspondances modulaires. Int. Math. Res. Not. 56, 3479–3510 (2005)

  10. 10

    Chazottes J.-R., Gouëzel S.: On almost-sure versions of classical limit theorems for dynamical systems. Probab. Theory Related Fields 138(1–2), 195–234 (2007)

  11. 11

    Chernov N.I.: Limit theorems and Markov approximations for chaotic dynamical systems. Probab. Theory Related Fields 101(3), 321–362 (1995)

  12. 12

    Denker, M.: The central limit theorem for dynamical systems, Dynamical systems and ergodic theory (Warsaw, 1986), pp. 33–62. Banach Center Publ., 23, PWN, Warsaw (1989)

  13. 13

    Denker M., Philipp W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergodic Theory Dyn. Syst. 4(4), 541–552 (1984)

  14. 14

    Denker M., Przytycki F., Urbański M.: On the transfer operator for rational functions on the Riemann sphere. Ergodic Theory Dyn. Syst. 16(2), 255–266 (1996)

  15. 15

    Dinh T.C., Dupont C.: Dimension de la mesure d’équilibre d’applications méromorphes. J. Geom. Anal. 14(4), 613–627 (2004)

  16. 16

    Dinh T.C., Nguyen V.A., Sibony N.: On thermodynamics of rational maps on the Riemann sphere. Ergodic Theory Dyn. Syst. 27(4), 1095–1109 (2007)

  17. 17

    Dinh, T.C., Nguyen, V.A., Sibony, N.: Exponential estimates for plurisubharmonic functions and stochastic dynamics, preprint arXiv 0801.1983 (2008)

  18. 18

    Dinh T.C., Sibony N.: Decay of correlations and the central limit theorem for meromorphic maps. Comm. Pure Appl. Math. 59(5), 754–768 (2006)

  19. 19

    Dinh, T.C., Sibony, N.: Super-potentials of positive closed currents, intersection theory and dynamics. Acta Math. (to appear)

  20. 20

    Dolgopyat D.: Limit theorems for partially hyperbolic systems. Trans. Am. Math. Soc. 356(4), 1637–1689 (2004)

  21. 21

    Dupont C.: Formule de Pesin et applications méromorphes. Bull. Braz. Math. Soc. (N.S.) 37(3), 393–418 (2006)

  22. 22

    Fornaess, J.E., Sibony, N.: Complex dynamics in higher dimensions. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, Complex potential theory (Montreal, PQ, 1993), pp. 131–186. Kluwer, Dordrecht (1994)

  23. 23

    Gordin M.I.: The central limit theorem for stationary processes. Sov. Math. Dokl. 10, 1174–1176 (1969)

  24. 24

    Hofbauer F., Keller G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180(1), 119–140 (1982)

  25. 25

    Ibragimov I.A.: Some limit theorems for stationary processes. Theory Prob. Appl 7(4), 349–382 (1962)

  26. 26

    Lacey M., Philipp W.: A note on the almost sure central limit theorem. Stat. Probab. Lett. 9(3), 201–205 (1990)

  27. 27

    Ledrappier F.: Propriétés ergodiques des mesures de Sinaï. Inst. Hautes Études Sci. Publ. Math. 59, 163–188 (1984)

  28. 28

    Liverani, C.: Central limit theorem for deterministic systems, International Conference on Dynamical Systems (Montevideo, 1995). Pitman Res. Notes Math. Ser., 362 (1996)

  29. 29

    Lojasiewicz S.: Introduction to Complex Analytic Geometry. Birkhäuser, Basel (1991)

  30. 30

    Mañé R.: The Hausdorff dimension of invariant probabilities of rational maps. Lecture Notes in Mathematics, vol. 1331. Springer, Berlin (1998)

  31. 31

    Mattila P.: Geometry of Sets and Measures in Euclidian Spaces. Cambridge University Press, Cambridge (1995)

  32. 32

    Melbourne I., Nicol M.: Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260(1), 131–146 (2005)

  33. 33

    Pesin Ja.B.: Characteristic Lyapounoff exponents and smooth ergodic theory. Russ. Math. Surv. 32(4), 55–114 (1977)

  34. 34

    Pesin, Ja.B.: Dimension theory in dynamical systems. Chicago Lectures in Mathematics (1997)

  35. 35

    Philipp, W., Stout, W.: Almost sure invariance principles for partial sums of weakly dependent random variables. Mem. Amer. Math. Soc. 2(2), no. 161 (1975)

  36. 36

    Przytycki F., Urbański M., Zdunik A.: Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I. Ann. Math. (2) 130(1), 1–40 (1989)

  37. 37

    Ratner M.: The central limit theorem for geodesic flows on n-dimensional manifolds of negative curvature. Israel J. Math. 16, 181–197 (1973)

  38. 38

    Sibony, N.: Dynamique des applications rationnelles de \({\mathbb P^k}\). In: Dynamique et Géométrie Complexes, Panoramas et Synthèses No 8, SMF (1999)

  39. 39

    Sinai Y.G.: The central limit theorem for geodesic flows on manifolds of constant negative curvature. Sov. Math. Dokl. 1, 983–987 (1960)

  40. 40

    Zdunik A.: Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math. 99(3), 627–649 (1990)

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Correspondence to Christophe Dupont.

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Dupont, C. Bernoulli coding map and almost sure invariance principle for endomorphisms of \({\mathbb P^k}\) . Probab. Theory Relat. Fields 146, 337–359 (2010). https://doi.org/10.1007/s00440-008-0192-4

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  • Holomorphic dynamics
  • Bernoulli coding map
  • Almost sure invariance principle

Mathematics Subject Classification (2000)

  • 37F10
  • 37C40
  • 60F17