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

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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that for every rational map on the Riemann sphere , if for every -critical point whose forward trajectory does not contain any other critical point, the growth of is at least of order for an appropriate constant as , then . Here is the so-called essential, dynamical or hyperbolic dimension, is Hausdorff dimension of and is the minimal exponent for conformal measures on . If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of also coincides with . We prove ergodicity of every -conformal measure on assuming has one critical point , no parabolic, and . Finally for every -conformal measure on (satisfying an additional assumption), assuming an exponential growth of , we prove the existence of a probability absolutely continuous with respect to , -invariant measure. In the Appendix we prove also for every non-renormalizable quadratic polynomial with not in the main cardioid in the Mandelbrot set.

**[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 Liapunov exponents and absolute continuity for maps of the interval*, Ergodic Theory Dynam. Systems**3**(1983), no. 1, 13–46. MR**743027**, 10.1017/S0143385700001802**[CEbook]**Pierre Collet and Jean-Pierre Eckmann,*Iterated maps on the interval as dynamical systems*, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. MR**613981****[DPU]**Manfred Denker, Feliks Przytycki, and Mariusz Urbański,*On the transfer operator for rational functions on the Riemann sphere*, Ergodic Theory Dynam. Systems**16**(1996), no. 2, 255–266. MR**1389624**, 10.1017/S0143385700008804**[DU1]**M. Denker and M. Urbański,*On Sullivan’s conformal measures for rational maps of the Riemann sphere*, Nonlinearity**4**(1991), no. 2, 365–384. MR**1107011****[DU2]**M. Denker and M. Urbański,*The capacity of parabolic Julia sets*, Math. Z.**211**(1992), no. 1, 73–86. MR**1179780**, 10.1007/BF02571418**[Gu]**Miguel de Guzmán,*Differentiation of integrals in 𝑅ⁿ*, Lecture Notes in Mathematics, Vol. 481, Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón. MR**0457661****[Guckenheimer]**John Guckenheimer,*Sensitive dependence to initial conditions for one-dimensional maps*, Comm. Math. Phys.**70**(1979), no. 2, 133–160. MR**553966****[GJ]**John Guckenheimer and Stewart Johnson,*Distortion of 𝑆-unimodal maps*, Ann. of Math. (2)**132**(1990), no. 1, 71–130. MR**1059936**, 10.2307/1971501**[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]**Einar Hille,*Analytic function theory. Vol. II*, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR**0201608****[K]**A. Katok,*Lyapunov exponents, entropy and periodic orbits for diffeomorphisms*, Inst. Hautes Études Sci. Publ. Math.**51**(1980), 137–173. MR**573822****[L]**M. Lyubich,*Geometry of quadratic polynomials: moduli, rigidity and local connectivity*, Preprint SUNY at Stony Brook, IMS 1993/9.**[LMi]**Mikhail Lyubich and John Milnor,*The Fibonacci unimodal map*, J. Amer. Math. Soc.**6**(1993), no. 2, 425–457. MR**1182670**, 10.1090/S0894-0347-1993-1182670-0**[M]**Ricardo Mañé,*On a theorem of Fatou*, Bol. Soc. Brasil. Mat. (N.S.)**24**(1993), no. 1, 1–11. MR**1224298**, 10.1007/BF01231694**[N]**Tomasz Nowicki,*Some dynamical properties of 𝑆-unimodal maps*, Fund. Math.**142**(1993), no. 1, 45–57. MR**1207470****[McM]**Curtis T. McMullen,*Complex dynamics and renormalization*, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR**1312365****[NS]**Tomasz Nowicki and Sebastian van Strien,*Invariant measures exist under a summability condition for unimodal maps*, Invent. Math.**105**(1991), no. 1, 123–136. MR**1109621**, 10.1007/BF01232258**[Prado]**E. Prado,*Ergodicity of conformal measures for quadratic polynomials*, Manuscript, May 23, 1994.**[P1]**Feliks Przytycki,*Lyapunov characteristic exponents are nonnegative*, Proc. Amer. Math. Soc.**119**(1993), no. 1, 309–317. MR**1186141**, 10.1090/S0002-9939-1993-1186141-9**[P2]**V. P. Havin and N. K. Nikolski (eds.),*Linear and complex analysis. Problem book 3. Part II*, Lecture Notes in Mathematics, vol. 1574, Springer-Verlag, Berlin, 1994. MR**1334346****[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]**Feliks Przytycki, Mariusz Urbański, and Anna Zdunik,*Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps. I*, Ann. of Math. (2)**130**(1989), no. 1, 1–40. MR**1005606**, 10.2307/1971475**[R]**Mary Rees,*Positive measure sets of ergodic rational maps*, Ann. Sci. École Norm. Sup. (4)**19**(1986), no. 3, 383–407. MR**870689****[S]**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**, 10.1007/BFb0061443**[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]**Mariusz Urbański,*Rational functions with no recurrent critical points*, Ergodic Theory Dynam. Systems**14**(1994), no. 2, 391–414. MR**1279476**, 10.1017/S0143385700007926**[Y]**J.-Ch. Yoccoz, Talks on several conferences.

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Additional Information

**Feliks Przytycki**

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

Email:
feliksp@impan.impan.gov.pl

DOI:
https://doi.org/10.1090/S0002-9947-98-01890-X

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