×

zbMATH — the first resource for mathematics

Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials. (English) Zbl 0892.58063
Summary: 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 \(\text{HD}_{\text{ess}} (J)=\alpha_{0}=\text{HD} (J) \).
Here \(\text{HD}_{\text{ess}} (J)\) is the so-called essential, dynamical or hyperbolic dimension, \(\text{HD} (J)\) is the 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 \(\text{HD}(J)\).
We prove ergodicity of every \(\alpha\)-conformal measure on \(J\) assuming \(f\) has one critical point \(c\in J\), not 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 \(\text{HD}_{\text{ess}} (J)=\text{HD} (J) \) also for every non-renormalizable quadratic polynomial \(z\mapsto z^{2}+c\) with \(c\) not in the main cardioid in the Mandelbrot set.

MSC:
37F99 Dynamical systems over complex numbers
37A99 Ergodic theory
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
28A78 Hausdorff and packing measures
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. Bloch and M. Lyubich, Measurable dynamics of S-unimodal maps of the interval, Ann. Sci. Éc. Norm. Sup. (4) 24 (1991), 545-573. · Zbl 0790.58024
[2] 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. · Zbl 0532.28014 · doi:10.1017/S0143385700001802 · doi.org
[3] Pierre Collet and Jean-Pierre Eckmann, Iterated maps on the interval as dynamical systems, Progress in Physics, vol. 1, Birkhäuser, Boston, Mass., 1980. · Zbl 0458.58002
[4] 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. · Zbl 0852.46024 · doi:10.1017/S0143385700008804 · doi.org
[5] M. Denker and M. Urbański, On Sullivan’s conformal measures for rational maps of the Riemann sphere, Nonlinearity 4 (1991), no. 2, 365 – 384. · Zbl 0722.58028
[6] M. Denker and M. Urbański, The capacity of parabolic Julia sets, Math. Z. 211 (1992), no. 1, 73 – 86. · Zbl 0763.30009 · doi:10.1007/BF02571418 · doi.org
[7] Miguel de Guzmán, Differentiation of integrals in \?\(^{n}\), 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. · Zbl 0598.28006
[8] John Guckenheimer, Sensitive dependence to initial conditions for one-dimensional maps, Comm. Math. Phys. 70 (1979), no. 2, 133 – 160. · Zbl 0429.58012
[9] John Guckenheimer and Stewart Johnson, Distortion of \?-unimodal maps, Ann. of Math. (2) 132 (1990), no. 1, 71 – 130. · Zbl 0708.58007 · doi:10.2307/1971501 · doi.org
[10] J. Graczyk and G. Świ[??]atek, Induced expansion for quadratic polynomials, Ann. Sci. Éc. Norm. Sup. (4) 29 (1996), 399-482. CMP 96:11
[11] J. Graczyk and G. Świ[??]atek, Holomorphic box mappings, Preprint IHES/M/1996/76.
[12] Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. · Zbl 0102.29401
[13] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 51 (1980), 137 – 173. · Zbl 0445.58015
[14] M. Lyubich, Geometry of quadratic polynomials: moduli, rigidity and local connectivity, Preprint SUNY at Stony Brook, IMS 1993/9.
[15] Mikhail Lyubich and John Milnor, The Fibonacci unimodal map, J. Amer. Math. Soc. 6 (1993), no. 2, 425 – 457. · Zbl 0778.58040
[16] Ricardo Mañé, On a theorem of Fatou, Bol. Soc. Brasil. Mat. (N.S.) 24 (1993), no. 1, 1 – 11. · Zbl 0781.30023 · doi:10.1007/BF01231694 · doi.org
[17] Tomasz Nowicki, Some dynamical properties of \?-unimodal maps, Fund. Math. 142 (1993), no. 1, 45 – 57. · Zbl 0821.58025
[18] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. · Zbl 0807.30013
[19] Tomasz Nowicki and Sebastian van Strien, Invariant measures exist under a summability condition for unimodal maps, Invent. Math. 105 (1991), no. 1, 123 – 136. · Zbl 0736.58030 · doi:10.1007/BF01232258 · doi.org
[20] E. Prado, Ergodicity of conformal measures for quadratic polynomials, Manuscript, May 23, 1994.
[21] Feliks Przytycki, Lyapunov characteristic exponents are nonnegative, Proc. Amer. Math. Soc. 119 (1993), no. 1, 309 – 317. · Zbl 0787.58037
[22] V. P. Havin and N. K. Nikolski , Linear and complex analysis. Problem book 3. Part I, Lecture Notes in Mathematics, vol. 1573, Springer-Verlag, Berlin, 1994. V. P. Havin and N. K. Nikolski , Linear and complex analysis. Problem book 3. Part II, Lecture Notes in Mathematics, vol. 1574, Springer-Verlag, Berlin, 1994. · Zbl 0893.30036
[23] -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”, F. Ledrappier, J. Lewowicz, S. Newhouse, Pitman Res. Notes in Math. 362, Longman (1996), 167-181. · Zbl 0868.58063
[24] F. Przytycki and M. Urbański, To appear.
[25] 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. · Zbl 0703.58036 · doi:10.2307/1971475 · doi.org
[26] Mary Rees, Positive measure sets of ergodic rational maps, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 383 – 407. · Zbl 0611.58038
[27] Dennis Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725 – 752. · doi:10.1007/BFb0061443 · doi.org
[28] M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia set, Preprint SUNY at Stony Brook, IMS 1991/7.
[29] Mariusz Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), no. 2, 391 – 414. · Zbl 0807.58025 · doi:10.1017/S0143385700007926 · doi.org
[30] J.-Ch. Yoccoz, Talks on several conferences.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.