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Transactions of the American Mathematical Society

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ISSN 1088-6850 (online) ISSN 0002-9947 (print)

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Ergodic and Bernoulli properties of analytic maps of complex projective space
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by Lorelei Koss PDF
Trans. Amer. Math. Soc. 354 (2002), 2417-2459 Request permission

Abstract:

We examine the measurable ergodic theory of analytic maps $F$ of complex projective space. We focus on two different classes of maps, Ueda maps of ${\mathbb P}^{n}$, and rational maps of the sphere with parabolic orbifold and Julia set equal to the entire sphere. We construct measures which are invariant, ergodic, weak- or strong-mixing, exact, or automorphically Bernoulli with respect to these maps. We discuss topological pressure and measures of maximal entropy ($h_{\mu }(F) = h_{top}(F)= \log (\deg F)$). We find analytic maps of ${\mathbb P}^1$ and ${\mathbb P}^2$ which are one-sided Bernoulli of maximal entropy, including examples where the maximal entropy measure lies in the smooth measure class. Further, we prove that for any integer $d>1$, there exists a rational map of the sphere which is one-sided Bernoulli of entropy $\log d$ with respect to a smooth measure.
References
  • Jonathan Aaronson, Michael Lin, and Benjamin Weiss, Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products, Israel J. Math. 33 (1979), no. 3-4, 198–224 (1980). A collection of invited papers on ergodic theory. MR 571530, DOI 10.1007/BF02762161
  • Roy L. Adler, L. Wayne Goodwyn, and Benjamin Weiss, Equivalence of topological Markov shifts, Israel J. Math. 27 (1977), no. 1, 48–63. MR 437715, DOI 10.1007/BF02761605
  • L.M. Abramov and V.A. Rokhlin, The entropy of a skew product of measure-preserving transformations, Amer. Math. Soc Transl. Ser. 2 48, 255 – 265.
  • Jonathan Ashley, Brian Marcus, and Selim Tuncel, The classification of one-sided Markov chains, Ergodic Theory Dynam. Systems 17 (1997), no. 2, 269–295. MR 1444053, DOI 10.1017/S0143385797069745
  • Julia A. Barnes, Applications of noninvertible ergodic theory to rational maps of the sphere, Diss. Summ. Math. 1 (1996), no. 1-2, 49–53. MR 1420723
  • J. Barnes and L. Koss, One-sided Lebesgue Bernoulli maps of the sphere of degree $n^2$ and $2n^2$, Internat. J. Math. Math. Sci. 23 (2000) 383 – 392.
  • 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
  • Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
  • L. Böettcher, The principal convergence laws for iterates and their applications to analysis, IZV. FIz.-Mat. Obshch. pri Imper. Kazanskom Univ. 13 (1903), no. 1, 1 – 37; 14 (1904), nos. 3–4, 155 – 234.
  • Rufus Bowen, Bernoulli equilibrium states for Axiom A diffeomorphisms, Math. Systems Theory 8 (1974/75), no. 4, 289–294. MR 387539, DOI 10.1007/BF01780576
  • Henk Bruin and Jane Hawkins, Examples of expanding $C^1$ maps having no $\sigma$-finite invariant measure equivalent to Lebesgue, Israel J. Math. 108 (1998), 83–107. MR 1669400, DOI 10.1007/BF02783043
  • Lennart Carleson and Theodore W. Gamelin, Complex dynamics, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1230383, DOI 10.1007/978-1-4612-4364-9
  • M. Denker and M. Urbański, Ergodic theory of equilibrium states for rational maps, Nonlinearity 4 (1991), no. 1, 103–134. MR 1092887
  • Adrien Douady and John H. Hubbard, A proof of Thurston’s topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263–297. MR 1251582, DOI 10.1007/BF02392534
  • Stanley Eigen and Jane Hawkins, Examples and properties of nonexact ergodic shift measures, Indag. Math. (N.S.) 10 (1999), no. 1, 25–44. MR 1691467, DOI 10.1016/S0019-3577(99)80003-2
  • A.E. Eremenko and M. Yu. Lyubich, The dynamics of analytic transformations, Leningrad Math. J. 1 (1990), 563 – 634.
  • A. È. Erëmenko and M. Yu. Lyubich, The dynamics of analytic transformations, Algebra i Analiz 1 (1989), no. 3, 1–70 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 3, 563–634. MR 1015124
  • John Erik Fornæss, Dynamics in several complex variables, CBMS Regional Conference Series in Mathematics, vol. 87, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. MR 1363948
  • John Erik Fornæss and Nessim Sibony, Complex dynamics in higher dimension. I, Astérisque 222 (1994), 5, 201–231. Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). MR 1285389
  • 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
  • M. Gromov, Entropy, homology and semialgebraic geometry, Astérisque 145-146 (1987), 5, 225–240. Séminaire Bourbaki, Vol. 1985/86. MR 880035
  • 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
  • S. Ito, On the fractal curves induced from the complex radix expansion, Tokyo J. Math. 12 (1989), 299 – 320.
  • Shunji Ito and Makoto Ohtsuki, On the fractal curves induced from endomorphisms on a free group of rank $2$, Tokyo J. Math. 14 (1991), no. 2, 277–304. MR 1138167, DOI 10.3836/tjm/1270130372
  • Steven Arthur Kalikow, $T,\,T^{-1}$ transformation is not loosely Bernoulli, Ann. of Math. (2) 115 (1982), no. 2, 393–409. MR 647812, DOI 10.2307/1971397
  • Michael Keane, Strongly mixing $g$-measures, Invent. Math. 16 (1972), 309–324. MR 310193, DOI 10.1007/BF01425715
  • Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960, DOI 10.1007/978-1-4612-4752-4
  • S. Làttes, Sur l’iteration des substitutions rationelles et les fonctions de Poincarè, C.R. Acad. Sci. Paris 166 Ser. 1 Math. (1919), 26 – 28.
  • M. Yu. Lyubich, The dynamics of rational transforms: the topological picture, Russian Math. Surveys 41 (1986), 43 – 117.
  • 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
  • Ricardo Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), no. 1, 27–43. MR 736567, DOI 10.1007/BF02584743
  • Ricardo Mañé, Ergodic theory and differentiable dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 8, Springer-Verlag, Berlin, 1987. Translated from the Portuguese by Silvio Levy. MR 889254, DOI 10.1007/978-3-642-70335-5
  • Ricardo Mañé, On the Bernoulli property for rational maps, Ergodic Theory Dynam. Systems 5 (1985), no. 1, 71–88. MR 782789, DOI 10.1017/S0143385700002765
  • Mike Hurley, On topological entropy of maps, Ergodic Theory Dynam. Systems 15 (1995), no. 3, 557–568. MR 1336706, DOI 10.1017/S014338570000852X
  • Curt McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. (2) 125 (1987), no. 3, 467–493. MR 890160, DOI 10.2307/1971408
  • J. Milnor, Pasting together Julia sets — a worked out example of mating, preprint 1997.
  • Donald S. Ornstein, Factors of Bernoulli shifts, Israel J. Math. 21 (1975), no. 2-3, 145–153. MR 382599, DOI 10.1007/BF02760792
  • William Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products, Ergodic Theory Dynam. Systems 16 (1996), no. 3, 519–529. MR 1395050, DOI 10.1017/S0143385700008944
  • William Parry and Selim Tuncel, Classification problems in ergodic theory, Statistics: Textbooks and Monographs, vol. 41, Cambridge University Press, Cambridge-New York, 1982. MR 666871
  • Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
  • V.A. Rokhlin, Exact endomorphisms of a Lebesgue Space, Amer. Math. Soc. Transl. Ser. 2 39 (1964), 1–36.
  • V.A. Rokhlin, On the fundamental ideas of measure theory, Amer. Math. Soc. Transl. Ser. 1 71 (1952), 1–55.
  • D. Ruelle, Statistical mechanics of a one-dimensional lattice gas, Comm. Math. Phys. 9 (1968), 267–278. MR 234697
  • W. Thurston, Lecture Notes, CBMS Conference, University of Minnesota at Duluth, 1983.
  • T. Ueda, Complex dynamical systems on projective spaces, Surikaisekikenkyusho Kokyuroka no. 814 (1992), 169 – 186.
  • Tetsuo Ueda, Critical orbits of holomorphic maps on projective spaces, J. Geom. Anal. 8 (1998), no. 2, 319–334. MR 1705160, DOI 10.1007/BF02921645
  • Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag, New York-Berlin, 1982. MR 648108
  • P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121 – 153.
  • Peter Walters, Some results on the classification of non-invertible measure preserving transformations, Recent advances in topological dynamics (Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Math., Vol. 318, Springer, Berlin, 1973, pp. 266–276. MR 0393424
  • Anna Zdunik, Parabolic orbifolds and the dimension of the maximal measure for rational maps, Invent. Math. 99 (1990), no. 3, 627–649. MR 1032883, DOI 10.1007/BF01234434
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Additional Information
  • Lorelei Koss
  • Affiliation: Department of Mathematics and Computer Science, Dickinson College, P.O. Box 1773, Carlisle, Pennsylvania 17013
  • MR Author ID: 662937
  • Email: koss@dickinson.edu
  • Received by editor(s): March 22, 1999
  • Received by editor(s) in revised form: March 14, 2000
  • Published electronically: February 7, 2002
  • Additional Notes: Supported in part by GAANN (Graduate Assistance in Areas of National Need) Fellowship
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2417-2459
  • MSC (2000): Primary 37A25, 37A35, 37F10
  • DOI: https://doi.org/10.1090/S0002-9947-02-02725-3
  • MathSciNet review: 1885659