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When do two rational functions
have the same Julia set?

Authors: G. Levin and F. Przytycki
Journal: Proc. Amer. Math. Soc. 125 (1997), 2179-2190
MSC (1991): Primary 58F23
MathSciNet review: 1376996
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Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that non-exceptional rational functions $f$ and $g$ on the Riemann sphere have the same measure of maximal entropy iff there exist iterates $F$ of $f$ and $G$ of $g$ and natural numbers $M,N$ such that

\begin{equation*}(G^{-1}\circ G)\circ G^{M} = (F^{-1}\circ F) \circ F^{N}.\tag {$*$} \end{equation*}

If one assumes only that $f,g$ have the same Julia set and no singular or parabolic domains of normality for the iterates, one also proves $(*)$.

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  • [B] A. F. Beardon, Iteration of rational functions, Graduate Texts in Mathematics 132, Springer-Verlag, 1991. MR 92j:30026
  • [B1] A. F. Beardon, Symmetries of Julia sets, Bull. London Math. Soc. 22 (1990), 576-582. MR 92f:30033
  • [B2] A. F. Beardon., Polynomials with identical Julia sets, Complex Variables 17 (1992), 195-200. MR 93k:30033
  • [BE] I. N. Baker, A. Eremenko, A problem on Julia sets, Ann. Acad. Sci. Fenn. 12 (1987), 229-236. MR 89g:30047
  • [CG] L. Carleson and W. Gamelin, Complex dynamics, Springer-Verlag, 1993. MR 94h:30033
  • [DH] A. Douady, J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), 263-297. MR 94j:58143
  • [E] A. Eremenko, On some functional equations connected with iteration of rational functions, Algebra i Analiz (Leningrad Math. J.) 1.4 (1989), 102-116. MR 90m:30030
  • [ELyu] A. Eremenko, M. Lyubich, Dynamics of Analytic transformations, Algebra i Analiz (Leningrad J. Math.) 1.3 (1989), 1-70. MR 91b:58109
  • [F1] P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Mat. France 47 (1919), 161-271; 48 (1920), 33-94, 208-314.
  • [F2] P. Fatou, Sur les fonctions qui admettent plusieurs théorèmes de multiplication, C.R.A.S. 173 (1921), 571-573; Sur l'itération analytique et les substitutions permutables., J. de Math. 2 (1923), 343.
  • [Fe] J. Fernandez, A note on the Julia set for polynomials, Complex Variables 12 (1989), 83-85. MR 91b:30069
  • [FLM] A. Freire, A. Lopes, R. Mañé, An invariant measure for rational maps, Bol. Soc. Bras. Math. 14.1 (1983), 45-62. MR 85m:58110b
  • [J1] G. Julia, Mémoire sur l'itération des fonctions rationnelles, J. Math. Pure Appl. 8 (1918), 47-245.
  • [J2] G. Julia, Mémoire sur la permutabilité des fractions rationnelles, Ann. Ecole Norm. Sup. 39 (1922), 131-215.
  • [L] G. M. Levin, On symmetries on a Julia set, Mat. Zametki 48.5 (1990), 72-79 (in Russian); Math. Notes 48.5-6 (1991), 1126-1131. MR 92e:30015
  • [Led1] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergod. Th. & Dynam. Sys. 1 (1981), 77-93. MR 82k:28018
  • [Led2] F. Ledrappier, Quelques propriétés ergodiques des applications rationnelles, C.R.A.S.P. 299 (1984), 37-40. MR 86c:58091
  • [Lyu] M. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergod. Th. & Dynam. Sys. 3 (1983), 351-386. MR 85k:58049
  • [M1] R. Mañé, On the uniqueness of the maximizing measure for rational maps, Bol. Soc. Bras. Math. 14.1 (1983), 27-43. MR 85m:58110a
  • [M2] R. Mañé, The Hausdorff dimension of invariant probabilities of rational maps, Dynamical Systems Valparaiso 1986, L. N. Math. 1331, Springer. MR 90j:58073
  • [Pa] W. Parry, Entropy and Generators in Ergodic Theory, W. A. Benjamin, Inc., New York, 1969. MR 41:7071
  • [Pe] Ja. B. Pesin, Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys 32 (1977), 55-114.
  • [P1] F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of attractive basin for a holomorphic map, Invent. Math. 80 (1985), 161-179. MR 86g:30035
  • [P2] F. Przytycki, Riemann map and holomorphic dynamics, Invent. Math. 85 (1986), 439-455. MR 88j:58058
  • [P3] F. Przytycki, On measure and Hausdorff dimension of Julia sets for holomorphic Collet-Eckmann maps, in International Conference on Dynamical Systems, Montevideo 1995, a tribute to Ricardo Mañé. Pitman Research Notes in Mathematics 362.
  • [PSV] F. Przytycki, J. Skrzypczak, A. Volberg, The dichotomy for the boundary of a parabolic simply-connected basin, A manuscript, Spring 1995.
  • [PZ] F. Przytycki, A. Zdunik, Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps, geometric coding trees technique, Fund. Math. 145 (1994), 65-77. MR 95k:30054
  • [R1] J. F. Ritt, Permutable rational functions, Trans. Amer. Math. Soc. 25 (1923), 399-448.
  • [R2] J. F. Ritt, Periodic functions with a multiplication theorem, Trans. Amer. Math. Soc. 23 (1922), 16-25.
  • [Ro] V. A. Rohlin, Lectures on the entropy theory of transformations with invariant measures, Usp. Mat. Nauk 22.5 (1967), 3-56 (in Russian); Russ. Math. Surv. 22.5 (1967), 1-52. MR 36:349
  • [Ru] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Mat. 9 (1978), 83-87. MR 80f:58026
  • [S] D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Annals of Mathematics 122 (1985), 401-418. MR 87i:58103
  • [T] W. Thurston, On combinatorics of iterated rational maps, Preprint, 1985.

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Additional Information

G. Levin
Affiliation: Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel

F. Przytycki
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950 Warsaw, Poland

Received by editor(s): January 27, 1995
Received by editor(s) in revised form: February 9, 1996
Additional Notes: The preprint version of this paper has the title Rational maps, common Julia sets, functional equations.
Communicated by: Mary Rees
Article copyright: © Copyright 1997 American Mathematical Society

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