Attractors for graph critical rational maps
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- by Alexander Blokh and Michał Misiurewicz PDF
- Trans. Amer. Math. Soc. 354 (2002), 3639-3661 Request permission
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 $f$.References
- Julia A. Barnes, Conservative exact rational maps of the sphere, J. Math. Anal. Appl. 230 (1999), no. 2, 350–374. MR 1672223, DOI 10.1006/jmaa.1998.6213
- Alexander M. Blokh, The “spectral” decomposition for one-dimensional maps, Dynamics reported, Dynam. Report. Expositions Dynam. Systems (N.S.), vol. 4, Springer, Berlin, 1995, pp. 1–59. MR 1346496
- S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479–494. MR 0000088
- A. Blokh and G. Levin, Growing trees, laminations and the dynamics on the Julia set, Ergodic Theory Dynam. Systems (to appear).
- Preben Alstrøm, Dimitris Stassinopoulos, and H. Eugene Stanley, Images and distributions obtained from affine transformations, Phys. Rev. A (3) 41 (1990), no. 10, 5290–5293. MR 1055287, DOI 10.1103/PhysRevA.41.5290
- A. M. Blokh and M. Yu. Lyubich, Ergodicity of transitive unimodal transformations of the interval, Ukrain. Mat. Zh. 41 (1989), no. 7, 985–988, 1008 (Russian); English transl., Ukrainian Math. J. 41 (1989), no. 7, 841–844 (1990). MR 1024300, DOI 10.1007/BF01060708
- A. M. Blokh and M. Yu. Lyubich, Decomposition of one-dimensional dynamical systems into ergodic components. The case of a negative Schwarzian derivative, Algebra i Analiz 1 (1989), no. 1, 128–145 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 137–155. MR 1015337
- A. M. Blokh and M. Yu. Lyubich, Measurable dynamics of $S$-unimodal maps of the interval, Ann. Sci. École Norm. Sup. (4) 24 (1991), no. 5, 545–573. MR 1132757
- A. Blokh, J. Mayer and L. Oversteegen, Recurrent critical points and typical limit sets for conformal measures, Topology Appl. 108 (2000), 233–244.
- Alexander Blokh and MichałMisiurewicz, Wild attractors of polymodal negative Schwarzian maps, Comm. Math. Phys. 199 (1998), no. 2, 397–416. MR 1666867, DOI 10.1007/s002200050506
- Rufus Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics, Vol. 470, Springer-Verlag, Berlin-New York, 1975. MR 0442989
- M. Denker, R. D. Mauldin, Z. Nitecki, and M. Urbański, Conformal measures for rational functions revisited, Fund. Math. 157 (1998), no. 2-3, 161–173. Dedicated to the memory of Wiesław Szlenk. MR 1636885, DOI 10.4064/fm_{1}998_{1}57_{2}-3_{1}_{1}61_{1}73
- 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), no. 4, 431–444. Hyperbolic behaviour of dynamical systems (Paris, 1990). MR 1096102
- M. Yu. Lyubich, Typical behavior of trajectories of the rational mapping of a sphere, Dokl. Akad. Nauk SSSR 268 (1983), no. 1, 29–32 (Russian). MR 687919
- —, Ergodic theory for smooth one-dimensional dynamical systems, SUNY at Stony Brook, Preprint #1991/11 (1991).
- Mikhail Lyubich and Yair Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), no. 1, 17–94. MR 1601430
- Ricardo Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 1, 1–11. MR 1224298, DOI 10.1007/BF01231694
- M. Martens, W. de Melo, and S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992), no. 3-4, 273–318. MR 1161268, DOI 10.1007/BF02392981
- Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
- Curtis T. McMullen, Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), no. 4, 535–593. MR 1789177, DOI 10.1007/s000140050140
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- J. Milnor, Dynamics in one complex variable, Friedr. Vieweg and Sohn, Braunschweig-Wiesbaden, 1999.
- Eduardo A. Prado, Ergodicity of conformal measures for unimodal polynomials, Conform. Geom. Dyn. 2 (1998), 29–44. MR 1613051, DOI 10.1090/S1088-4173-98-00019-8
- Feliks Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), no. 5, 2081–2099. MR 1615954, DOI 10.1090/S0002-9947-99-02195-9
- Dennis Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725–752. MR 730296, DOI 10.1007/BFb0061443
- Ya. B. Pesin, A generalization of Carathéodory’s construction for dimensional characteristic of dynamic systems, Statistical physics and dynamical systems (Köszeg, 1984) Progr. Phys., vol. 10, Birkhäuser Boston, Boston, MA, 1985, pp. 191–201. MR 821297
- 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, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. Math. Soc. 236 (1978), 121–153. MR 466493, DOI 10.1090/S0002-9947-1978-0466493-1
Additional Information
- Alexander Blokh
- Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
- MR Author ID: 196866
- Email: ablokh@math.uab.edu
- Michał Misiurewicz
- Affiliation: Department of Mathematical Sciences, Indiana University - Purdue University Indianapolis, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
- MR Author ID: 125475
- Email: mmisiure@math.iupui.edu
- 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 - © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 3639-3661
- MSC (2000): Primary 37F10; Secondary 37E25
- DOI: https://doi.org/10.1090/S0002-9947-02-02999-9
- MathSciNet review: 1911515