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We use Beurling estimates and Zdunik’s theorem to prove that the support of a lamination of the circle corresponding to a connected polynomial Julia set has zero length, unless f is conjugate to a Chebyshev polynomial. Equivalently, except for the Chebyshev case, the biaccessible points in the connected polynomial Julia set have zero harmonic measure. A connected, locally connected, full, compact subset K of the complex plane C can be topologically described by a lamination of a circle, which tells how to pinch the circle to obtain K (see, e.g., [Dou]). A lamination is an equivalence relation on the unit circle T, identifying points ζ and ζ ′ if they are mapped to one point in K by the Riemann uniformization map of the complement of K. To obtain a topological model of the compact set K, we glue together points of the unit circle, belonging to one equivalence class. The support of a lamination is defined as the union of all non-trivial (containing 2 or more points) equivalence classes, i.e. it includes those points which are identified with some other points. The laminations so defined are topologically fully characterized (among all equivalence relations on T, see [Dou]) by the following properties: (1) the graph {(ζ, ζ ) : ζ ∼ ζ ′} is a closed set in T× T, (2) the convex hulls of different equivalence classes are disjoint, (3) each equivalence class is totally disconnected. There are also analytical properties (e.g. the logarithmic capacity of each equivalence class is zero) which are not fully understood, and it is a difficult open question how to characterize laminations analytically among all equivalence relations on T. It also makes sense to consider laminations corresponding to not necessarily locally connected compacta, but those lamina2000 Mathematics Subject Classification: Primary 37F20; Secondary 30C85, 30D05.

On supports of dynamical laminations and biaccessible points in polynomial Julia sets