Ergodic Theory and Dynamical Systems



An inequality for laminations, Julia sets and ‘growing trees’


A. BLOKH a1 and G. LEVIN a2
a1 Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, AL 35294-2060, USA (e-mail: ablokh@math.uab.edu)
a2 Institute of Mathematics, Hebrew University, Givat Ram 91904, Jerusalem, Israel (e-mail: levin@math.huji.ac.il)

Abstract

For a closed lamination on the unit circle invariant under z\mapsto z^d we prove an inequality relating the number of points in the ‘gaps’ with infinite pairwise disjoint orbits to the degree; in particular, this gives estimates on the cardinality of any such ‘gap’ as well as on the number of distinct grand orbits of such ‘gaps’. As a tool, we introduce and study a dynamically defined growing tree in the quotient space. We also use our techniques to obtain for laminations an analog of Sullivan's no wandering domain theorem. Then we apply these results to Julia sets of polynomials.

(Received June 6 2000)
(Revised February 5 2001)