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

Feliks Przytycki

Fundamenta Mathematicae (1994)

  • Volume: 144, Issue: 3, page 259-278
  • ISSN: 0016-2736

Abstract

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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.

How to cite

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Przytycki, Feliks. "Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps." Fundamenta Mathematicae 144.3 (1994): 259-278. <http://eudml.org/doc/212028>.

@article{Przytycki1994,
abstract = {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.},
author = {Przytycki, Feliks},
journal = {Fundamenta Mathematicae},
keywords = {basin of attraction; invariant measures; holomorphic maps; Lyapunov exponents; accessibility; iteration},
language = {eng},
number = {3},
pages = {259-278},
title = {Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps},
url = {http://eudml.org/doc/212028},
volume = {144},
year = {1994},
}

TY - JOUR
AU - Przytycki, Feliks
TI - Accessibility of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps
JO - Fundamenta Mathematicae
PY - 1994
VL - 144
IS - 3
SP - 259
EP - 278
AB - 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.
LA - eng
KW - basin of attraction; invariant measures; holomorphic maps; Lyapunov exponents; accessibility; iteration
UR - http://eudml.org/doc/212028
ER -

References

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  13. [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. Zbl0569.58024
  14. [P2] F. Przytycki, Riemann map and holomorphic dynamics, ibid. 85 (1986), 439-455. Zbl0616.58029
  15. [P3] F. Przytycki, On invariant measures for iterations of holomorphic maps, in: Problems in Holomorphic Dynamic, preprint IMS 1992/7, SUNY at Stony Brook. 
  16. [P4] F. Przytycki, Polynomials in hyperbolic components, manuscript, Stony Brook 1992. 
  17. [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. Zbl0704.30035
  18. [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. Zbl0703.58036
  19. [R] D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), 83-87. Zbl0432.58013

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