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On explicit connections between dynamical and parameter spaces. (English) Zbl 1073.37054

Let \(M_\ell\) denote the Mandelbrot set for the family of polynomials \(f_c: S^2\to S^2\) of the form \(f_c(z)= z^\ell+ c\) of the Riemann sphere and \(K_c\) the filled in Julia set of \(f_c\), where \(\ell\) is fixed. It is shown that there exists a constant \(L_*\) such that \[ \text{dist}(c,\partial M_\ell)\leq L_*\ell^2\int_{K_c} |z|^{l-2} dz, \] where \(dz\) denotes the Lebesgue measure on the plane. This result is then refined for hyperbolic and nonhyperbolic components of \(M_\ell\).
The second result of the paper deals with the velocity of motion of periodic orbits. Let \(b_1,\dots, b_n\) be an orbit of \(f_c\) of prime period and with multiplier \(\rho\not\in\{0,1\}\). Since \(\rho\neq 1\) there is a holomorphic extension of the orbit to a periodic orbit (of the same period) of \(f_{\widehat c}\) in some neighbourhood of \(c\) with multiplier \(\rho(\widehat c)\), which is a holomorphic function. Defining \[ A(z)= \sum^n_{k=1} {1\over (z- b_k)^2}+ {(f^n_c)''(b_k)\over \rho(1- \rho)(z- b_k)}, \] the relation between the parameter plane (represented by the derivative \(\rho'\)) and the dynamical plane is given by \[ A(z)= (TA)(z)+ {\rho'(c)\over \rho(c)(z- c)}. \] Here, \(T\) denotes the transfer operator.
Many applications and corollaries are given (including the Douady-Hubbard-Sullivan theorem). The paper concludes with an extension of the results to polynomial-like mappings.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
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[1] Ahlfors, L., Lectures on Quasiconformal Mappings (1966), New York: Van Nostrand, New York · Zbl 0138.06002
[2] Carleson, L.; Gamelin, T., Complex Dynamics (1993), Berlin: Springer-Verlag, Berlin · Zbl 0782.30022
[3] Coullet, P.; Tresser, C., Itération d’endomorphismes et groupe de renormalisation, C.R. Acad. Sci. Paris, 287 A, 577-580 (1978) · Zbl 0402.54046
[4] A. Douady,Systèmes dynamiques holomorphes, Seminare Bourbaki, Vol. 1982-83, no. 599; Asterisque105-106 (1983), 39-63 · Zbl 0532.30019
[5] Douady, A.; Hubbard, J. H., Itération des polynômes quadratiques complexes, C.R. Acad. Sci. Paris, 294, 123-126 (1982) · Zbl 0483.30014
[6] A. Douady and J. H. Hubbard,Etude dynamique des polynômes complexes, Publ. Math. Orsay, 84-02 (1984), 84-04 (1985). · Zbl 0552.30018
[7] Douady, A.; Hubbard, J. H., On the dynamics of polynomial-like maps, Ann. Sci. École Norm. Sup., 18, 287-343 (1985) · Zbl 0587.30028
[8] A. Epstein,Infinitesimal Thurston rigidity and the Faton-Shishikura inequality, preprint, Stony Brook, 1999.
[9] Feigenbaum, M., Qualitative universality for a class of non-linear transformations, J. Statist. Phys., 19, 25-52 (1978) · Zbl 0509.58037 · doi:10.1007/BF01020332
[10] Hubbard, J. H., Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological Methods in Modern Mathematics, 467-511 (1993), Houston: Publish or Perish, Houston · Zbl 0797.58049
[11] Levin, G., On an analytic approach to the Fatou conjecture, Fund. Math., 171, 177-196 (2002) · Zbl 0984.37046
[12] Levin, G.; Sodin, M.; Yuditski, P., A Ruelle operator for a real Julia set, Comm. Math. Phys., 141, 119-132 (1991) · Zbl 0749.58039 · doi:10.1007/BF02100007
[13] Lyubich, M., Feigenbaum-Coullet-Tresser universality and Milnor’s hairiness conjecture, Ann. of Math., 149, 2, 319-420 (1999) · Zbl 0945.37012 · doi:10.2307/120968
[14] P. Makienko,Remarks on Ruelle operator and invariant line field problem. Part I, preprint FIM, 25, Zurich, July 2000;Part II, preprint 2001.
[15] Mane, R.; Sad, P.; Sullivan, D., On the dynamics of rational maps, Ann. Sci. École Norm. Sup., 16, 193-217 (1983) · Zbl 0524.58025
[16] C. McMullen,Complex Dynamics and Renormalization, Ann. of Math. Stud.135, Princeton University Press, 1994. · Zbl 0822.30002
[17] C. McMullen,Renormalization and 3-Manifolds which Fiber over the Circle, Ann. of Math. Stud.142. Princeton University Press, 1996. · Zbl 0860.58002
[18] Rivera-Letelier, J., On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets, Fund. Math., 170, 287-317 (2001) · Zbl 0985.37041 · doi:10.4064/fm170-3-6
[19] D. Sullivan,Bounds, quadratic differentials and renormalization conjecture, inMathematica into the Twenty-First Century, AMS Centennial Publications, 1991.
[20] Tsujii, M., A simple proof for monotonicity of entropy in the quadratic family, Ergodic Theory Dynam. Systems, 20, 925-933 (2000) · Zbl 0957.37008 · doi:10.1017/S014338570000050X
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