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

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

MathSciNet review:
1407501

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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 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*, 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 -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.

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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