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Attractors for graph critical rational maps

Authors: Alexander Blokh and Michal Misiurewicz
Journal: Trans. Amer. Math. Soc. 354 (2002), 3639-3661
MSC (2000): Primary 37F10; Secondary 37E25
Published electronically: April 30, 2002
MathSciNet review: 1911515
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Abstract: We call a rational map $f$ graph critical if any critical point either belongs to an invariant finite graph $G$, or has minimal limit set, or is non-recurrent and has limit set disjoint from $G$. We prove that, for any conformal measure, either for almost every point of the Julia set $J(f)$ its limit set coincides with $J(f)$, or for almost every point of $J(f)$ its limit set coincides with the limit set of a critical point of $f$.

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  • [Ba] J. Barnes, Conservative exact rational maps of the sphere, J. Math. Anal. Appl. 230 (1999), 350-374. MR 2000d:37048
  • [B1] A. Blokh, The spectral decomposition for one-dimensional maps, Dynamics Reported 4 (1995), 1-59. MR 96e:58087
  • [B2] -, Dynamical systems on one-dimensional branched manifolds. I, J. Soviet Math. 48 (1990), 500-508; II,, vol. 48, 1990, pp. 668-674; III,, vol. 49, 1990, pp. 875-883. MR 88j:58053; MR 89i:58056; MR 89i:58057
  • [BLe] A. Blokh and G. Levin, Growing trees, laminations and the dynamics on the Julia set, Ergodic Theory Dynam. Systems (to appear).
  • [BL1] A. Blokh and M. Lyubich, Attractors of maps of the interval, Dynamical Systems and Ergodic Theory (Warsaw, 1986), Banach Center Publ., vol. 23, PWN, Warsaw, 1989, pp. 427-442. MR 91c:58068
  • [BL2] -, Ergodicity of transitive unimodal transformations of the interval, Ukrainian Math. J. 41 (1989), 841-844. MR 90k:58109
  • [BL3] -, Decomposition of one-dimensional dynamical systems into ergodic components. The case of a negative Schwarzian derivative, Leningrad Math. Jour. 1 (1990), 137-155. MR 91d:58129
  • [BL4] -, Measurable dynamics of S-unimodal maps of the interval, Ann. Sci. Ecole Norm. Sup. (4) 24 (1991), 545-573. MR 93f:58132
  • [BMO] A. Blokh, J. Mayer and L. Oversteegen, Recurrent critical points and typical limit sets for conformal measures, Topology Appl. 108 (2000), 233-244.
  • [BM] A. Blokh and M. Misiurewicz, Wild attractors of polymodal negative Schwarzian maps, Comm. Math. Phys. 199 (1998), 397-416. MR 99k:58116
  • [Bo] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer, Berlin-New York, 1975. MR 56:1364
  • [DMNU] M. Denker, D. Mauldin, Z. Nitecki and M. Urbanski, Conformal measures for rational functions revisited, Fund. Math. 157 (1998), 161-173. MR 99j:58122
  • [GPS] P. Grzegorczyk, F. Przytycki and W. Szlenk, On iterations of Misiurewicz's rational maps on the Riemann sphere, Ann. Inst. H. Poincaré, Phys. Théor. 53 (1990), 431-444. MR 92d:30017
  • [L1] M. Lyubich, Typical behavior of trajectories of the rational mappings of a sphere, Soviet Math. Dokl. 27:1 (1983), 22-25. MR 84f:30036
  • [L2] -, Ergodic theory for smooth one-dimensional dynamical systems, SUNY at Stony Brook, Preprint #1991/11 (1991).
  • [LM] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), 17-94. MR 98k:58191
  • [Ma] R. Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), 1-11. MR 94g:58188
  • [MMS] M. Martens, W. de Melo and S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992), 273-318. MR 93d:58137
  • [McM1] C. T. McMullen, Complex dynamics and renormalization, Annals of Math. Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 96b:58097
  • [McM2] C. T. McMullen, Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps, Comm. Math. Helv. 75 (2000), 535-593. MR 2001m:37089
  • [M1] J. Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985), 177-195 (Correction and remarks: vol. 102 (1985), 517-519). MR 87i:58109
  • [M2] J. Milnor, Dynamics in one complex variable, Friedr. Vieweg and Sohn, Braunschweig-Wiesbaden, 1999.
  • [Pr] E. Prado, Ergodicity of conformal measures for unimodal polynomials, Conform. Geom. Dyn. 2 (1988), 29-44. MR 99g:58106
  • [P] F. Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), 2081-2099. MR 99h:58110
  • [Su1] D. Sullivan, Conformal dynamical systems, Geometric Dynamics, Lecture Notes in Math., vol. 1007, Springer, Berlin-New York, 1983, pp. 725-752. MR 85m:58112
  • [Su2] -, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Annals of Math. 122 (1985), 401-418. MR 87i:58102
  • [U] M. Urbanski, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414. MR 95g:58191
  • [W] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121-153. MR 57:6371

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Additional Information

Alexander Blokh
Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060

Michal Misiurewicz
Affiliation: Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216

Keywords: Complex dynamics, attractors, conformal measures, postcritical set
Received by editor(s): July 7, 2000
Received by editor(s) in revised form: December 20, 2001
Published electronically: April 30, 2002
Additional Notes: The first author was partially supported by NSF grant DMS 9970363
The second author was partially supported by NSF grant DMS 9970543
Article copyright: © Copyright 2002 American Mathematical Society

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