Recurrent critical points and typical limit sets for conformal measures

Academic Article

Abstract

  • For a rational f : ℂ̂ → ℂ̂ with a conformal measure μ we show that if there is a subset of the Julia set J(f) of positive μ-measure whose points are not eventual preimages of critical or parabolic points and have limit sets not contained in the union of the limit sets of recurrent critical points, then μ is non-atomic, μ(J(f)) = 1, ω(x) = J(f) for μ-a.e. point x ∈ J(f) and f is conservative, ergodic and exact. The proof uses a version of the Lebesgue Density Theorem valid for Borel measures and conformal balls. © 2000 Elsevier Science B.V. All rights reserved.
  • Published In

    Author List

  • Blokh AM; Mayer JC; Oversteegen LG
  • Start Page

  • 233
  • End Page

  • 244
  • Volume

  • 108
  • Issue

  • 3