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

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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.
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  • Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland
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