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## Fundamenta Mathematicae

1994 | 144 | 3 | 259-278
Tytuł artykułu

### Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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. $f^{-1}(A) = A$), 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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
259-278
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-04-30
poprawiono
1993-09-28
Twórcy
autor
• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
Bibliografia
• [D] A. Douady, informal talk at the Durham Symposium, 1988.
• [DH] A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. École Norm. Sup. (4) 18 (1985), 287-343.
• [EL] A. È. Eremenko and G. M. Levin, On periodic points of polynomials, Ukrain. Mat. Zh. 41 (1989), 1467-1471 (in Russian).
• [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.
• [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.
• [H1] M. Herman, Exemples de fractions rationnelles ayant une orbite dense sur la sphère de Riemann, Bull. Soc. Math. France 112 (1984), 93-142.
• [H2] M. Herman, Construction of some curious diffeomorphism of the Riemann sphere, J. London Math. Soc. 34 (1986), 375-384.
• [LevP] G. Levin and F. Przytycki, External rays to periodic points, preprint 24 (1992/93), the Hebrew University of Jerusalem.
• [LevS] G. Levin and M. Sodin, Polynomials with disconnected Julia sets and Green maps, preprint 23 (1990/1991), the Hebrew University of Jerusalem.
• [Mi1] J. Milnor, Dynamics in one complex variable: Introductory lectures, preprint IMS 1990/5, SUNY at Stony Brook.
• [Mi2] J. Milnor, Local connectivity of Julia sets: Expository lectures, preprint IMS 1992/11, SUNY at Stony Brook.
• [Pe] C. L. Petersen, On the Pommerenke-Levin-Yoccoz inequality, Ergodic Theory Dynamical Systems 13 (1993), 785-806.
• [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.
• [P2] F. Przytycki, Riemann map and holomorphic dynamics, ibid. 85 (1986), 439-455.
• [P3] F. Przytycki, On invariant measures for iterations of holomorphic maps, in: Problems in Holomorphic Dynamic, preprint IMS 1992/7, SUNY at Stony Brook.
• [P4] F. Przytycki, Polynomials in hyperbolic components, manuscript, Stony Brook 1992.
• [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.
• [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.
• [R] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), 83-87.
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