We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if A is completely invariant (i.e. ), and if μ is an arbitrary f-invariant measure with positive Lyapunov exponents on ∂A, then μ-almost every point q ∈ ∂A is accessible along a curve from A. In fact, we prove the accessibility of every “good” q, i.e. one for which “small neigh bourhoods arrive at large scale” under iteration of f. This generalizes the Douady-Eremenko-Levin-Petersen theorem on the accessibility of periodic sources. We prove a general “tree” version of this theorem. This allows us to deduce that on the limit set of a geometric coding tree (in particular, on the whole Julia set), if the diameters of the edges converge to 0 uniformly as the generation number tends to ∞, then every f-invariant probability ergodic measure with positive Lyapunov exponent is the image, via coding with the help of the tree, of an invariant measure on the full one-sided shift space. The assumption that f is holomorphic on A, or on the domain U of the tree, can be relaxed and one need not assume that f extends beyond A or U. Finally, we prove that if f is polynomial-like on a neighbourhood of ¯ℂ∖ A, then every “good” q ∈ ∂A is accessible along an external ray.
@article{bwmeta1.element.bwnjournal-article-fmv144i3p259bwm,
author = {Feliks Przytycki},
title = {Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps},
journal = {Fundamenta Mathematicae},
volume = {144},
year = {1994},
pages = {259-278},
zbl = {0812.58058},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv144i3p259bwm}
}
Przytycki, Feliks. Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps. Fundamenta Mathematicae, Tome 144 (1994) pp. 259-278. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv144i3p259bwm/
[00000] [D] A. Douady, informal talk at the Durham Symposium, 1988.
[00001] [DH] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. École Norm. Sup. (4) 18 (1985), 287-343. | Zbl 0587.30028
[00002] [EL] A. È. Eremenko and G. M. Levin, On periodic points of polynomials, Ukrain. Mat. Zh. 41 (1989), 1467-1471 (in Russian). | Zbl 0686.30019
[00003] [GM] L. R. Goldberg and J. Milnor, Fixed points of polynomial maps. Part II. Fixed point portraits, Ann. Sci. École Norm. Sup. (4) 26 (1993), 51-98. | Zbl 0771.30028
[00004] [GPS] P. Grzegorczyk, F. Przytycki and W. Szlenk, On iterations of Misiurewicz's rational maps on the Riemann sphere, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 431-444. | Zbl 0718.58031
[00005] [H1] M. Herman, Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), 93-142. | Zbl 0559.58020
[00006] [H2] M. Herman, Construction of some curious diffeomorphism of the Riemann sphere, J. London Math. Soc. 34 (1986), 375-384. | Zbl 0603.58017
[00007] [LevP] G. Levin and F. Przytycki, External rays to periodic points, preprint 24 (1992/93), the Hebrew University of Jerusalem.
[00008] [LevS] G. Levin and M. Sodin, Polynomials with disconnected Julia sets and Green maps, preprint 23 (1990/1991), the Hebrew University of Jerusalem.
[00009] [Mi1] J. Milnor, Dynamics in one complex variable: Introductory lectures, preprint IMS 1990/5, SUNY at Stony Brook.
[00010] [Mi2] J. Milnor, Local connectivity of Julia sets: Expository lectures, preprint IMS 1992/11, SUNY at Stony Brook.
[00011] [Pe] C. L. Petersen, On the Pommerenke-Levin-Yoccoz inequality, Ergodic Theory Dynamical Systems 13 (1993), 785-806. | Zbl 0802.30022
[00012] [P1] F. Przytycki, Hausdorff dimension of harmonic measure on the boundary of an attractive basin for a holomorphic map, Invent. Math. 80 (1985), 161-179. | Zbl 0569.58024
[00013] [P2] F. Przytycki, Riemann map and holomorphic dynamics, ibid. 85 (1986), 439-455. | Zbl 0616.58029
[00014] [P3] F. Przytycki, On invariant measures for iterations of holomorphic maps, in: Problems in Holomorphic Dynamic, preprint IMS 1992/7, SUNY at Stony Brook.
[00015] [P4] F. Przytycki, Polynomials in hyperbolic components, manuscript, Stony Brook 1992.
[00016] [PS] F. Przytycki and J. Skrzypczak, Convergence and pre-images of limit points for coding trees for iterations of holomorphic maps, Math. Ann. 290 (1991), 425-440. | Zbl 0704.30035
[00017] [PUZ] F. Przytycki, M. Urbański and A. Zdunik, Harmonic, Gibbs and Hausdorff measures for holomorphic maps, Part 1: Ann. of Math. 130 (1989), 1-40; Part 2: Studia Math. 97 (1991), 189-225. | Zbl 0703.58036
[00018] [R] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), 83-87. | Zbl 0432.58013