Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Iterations of holomorphic Collet-Eckmann maps:
conformal and invariant measures.
Appendix: On non-renormalizable
quadratic polynomials

Author: Feliks Przytycki
Journal: Trans. Amer. Math. Soc. 350 (1998), 717-742
MSC (1991): Primary 58F23
MathSciNet review: 1407501
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for every rational map on the Riemann sphere $f:\overline{\mathbb{C}} \to \overline{\mathbb{C}} $, if for every $f$-critical point $c\in J$ whose forward trajectory does not contain any other critical point, the growth of $|(f^{n})'(f(c))| $ is at least of order $\exp Q \sqrt n$ for an appropriate constant $Q$ as $n\to \infty $, then $\operatorname{HD}_{\operatorname {ess}} (J)=\alpha _{0}=\operatorname{HD} (J) $. Here $\operatorname{HD}_{\operatorname {ess}} (J)$ is the so-called essential, dynamical or hyperbolic dimension, $\operatorname{HD} (J)$ is Hausdorff dimension of $J$ and $\alpha _{0}$ is the minimal exponent for conformal measures on $J$. If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of $J$ also coincides with $\operatorname{HD}(J)$. We prove ergodicity of every $\alpha $-conformal measure on $J$ assuming $f$ has one critical point $c\in J$, no parabolic, and $\sum _{n=0}^{\infty }|(f^{n})'(f(c))|^{-1} <\infty $. Finally for every $\alpha $-conformal measure $\mu $ on $J$ (satisfying an additional assumption), assuming an exponential growth of $|(f^{n})'(f(c))|$, we prove the existence of a probability absolutely continuous with respect to $\mu $, $f$-invariant measure. In the Appendix we prove $\operatorname{HD}_{\operatorname {ess}} (J)=\operatorname {HD} (J) $ also for every non-renormalizable quadratic polynomial $z\mapsto z^{2}+c$ with $c$ not in the main cardioid in the Mandelbrot set.

References [Enhancements On Off] (What's this?)

  • [BL] M. Bloch and M. Lyubich, Measurable dynamics of S-unimodal maps of the interval, Ann. Sci. Éc. Norm. Sup. (4) 24 (1991), 545-573.
  • [CE] P. Collet and J.-P. Eckmann, Positive Lyapunov exponents and absolute continuity for maps of the interval, Ergodic Th. and Dyn. Sys. 3 (1983), 13-46. MR 85j:58092
  • [CEbook] -Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Basel-Boston-Stuttgart, 1980. MR 82j:58078
  • [DPU] M. Denker, F. Przytycki, and M. Urba\'{n}ski, On the transfer operator for rational functions on the Riemann sphere, Ergodic Th. and Dyn. Sys. 16 (1996), 255-266. MR 97e:58197
  • [DU1] M. Denker and M. Urba\'{n}ski, On Sullivan's conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), 365-384. MR 92f:58097
  • [DU2] -Capacity of parabolic Julia sets, Math. Zeit. 211 (1992), 73-86. MR 93j:30022
  • [Gu] M. de Guzmán, Differentiation of Integrals in $\mathbb{R}^{n}$, Lecture Notes in Math. vol. 481. Springer-Verlag, Berlin 1975. MR 56:15866
  • [Guckenheimer] J. Guckenheimer, Sensitive dependence on initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), 133-160. MR 82c:58037
  • [GJ] J. Guckenheimer and S. Johnson, Distortion of S-unimodal maps, Ann. of Math. 132 (1990), 71-130. MR 91g:58157
  • [GS1] J. Graczyk and G. \'{S}wi[??]atek, Induced expansion for quadratic polynomials, Ann. Sci. Éc. Norm. Sup. (4) 29 (1996), 399-482. CMP 96:11
  • [GS2] J. Graczyk and G. \'{S}wi[??]atek, Holomorphic box mappings, Preprint IHES/M/1996/76.
  • [H] E. Hille, Analytic Function Theory, Ginn and Company, Boston 1962. MR 34:1490
  • [K] A. Katok, Lyapunov exponents, entropy and periodic points for diffeomorphisms, Publ. Math. IHES 51 (1980), 137-173. MR 81i:28022
  • [L] M. Lyubich, Geometry of quadratic polynomials: moduli, rigidity and local connectivity, Preprint SUNY at Stony Brook, IMS 1993/9.
  • [LMi] M. Lyubich and J. Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), 425-457. MR 93h:58080
  • [M] R. Mañé, On a theorem of Fatou, Bol. Soc. Bras. Mat. 24 (1993), 1-12. MR 94g:58188
  • [N] T. Nowicki, Some dynamical properties of $S$-unimodal maps, Fund. Math. 142 (1993), 45-57. MR 94c:58111
  • [McM] C. McMullen, Complex Dynamics and Renormalization, Ann. of Math. Studies 135, Princeton University Press, 1994. MR 96b:58097
  • [NS] T. Nowicki and S. van Strien, Invariant measures exist under a summability condition for unimodal maps, Invent. Math. 105 (1991), 123-136. MR 93b:58094
  • [Prado] E. Prado, Ergodicity of conformal measures for quadratic polynomials, Manuscript, May 23, 1994.
  • [P1] F. Przytycki, Lyapunov characteristic exponents are non-negative, Proc. Amer. Math. Soc. 119(1) (1993), 309-317. MR 93k:58193
  • [P2] -Invariant measures for iterations of holomorphic maps, In "Linear and Complex Analysis. Problem Book 3" Part II, Eds. V. P. Havin, N. K. Nikolski. Lect. Notes in Math. 1574, Springer (1994), 450-454. MR 96c:00001b
  • [P3] -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'', (Eds) F. Ledrappier, J. Lewowicz, S. Newhouse, Pitman Res. Notes in Math. 362, Longman (1996), 167-181.
  • [PUbook] F. Przytycki and M. Urba\'{n}ski, To appear.
  • [PUZ] F. Przytycki, M. Urba\'{n}ski, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, I. Ann. of Math. 130 (1989), 1-40. MR 91i:58115
  • [R] M. Rees, Positive measure sets of ergodic rational maps, Ann. Sci. Éc. Norm. Sup. (4) 19 (1986), 383-407. MR 88g:58100
  • [S] D. Sullivan, Conformal dynamical systems, In "Geometric Dynamics", Lec. Notes in Math., vol. 1007, Springer, New York (1983), 725-752. MR 85m:58112
  • [Shi] M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia set, Preprint SUNY at Stony Brook, IMS 1991/7.
  • [U] M. Urba\'{n}ski, Rational functions with no recurrent critical points, Ergodic Th. and Dyn. Sys. 14.2 (1994), 391-414. MR 95g:58191
  • [Y] J.-Ch. Yoccoz, Talks on several conferences.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F23

Retrieve articles in all journals with MSC (1991): 58F23

Additional Information

Feliks Przytycki
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8 00 950 Warszawa, Poland

Received by editor(s): January 18, 1995
Received by editor(s) in revised form: July 28, 1995, and June 13, 1996
Additional Notes: The author acknowledges support by Polish KBN Grants 210469101 and 2 P301 01307 “Iteracje i Fraktale". He expresses also his gratitude to the Universities at Orleans and at Dijon in France, where parts of this paper were written
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society