Degree- @math -invariant laminations of the disk model the dynamical action of a degree- @math polynomial; such a lamination defines an equivalence relation on @math that corresponds to dynamical rays of an associated polynomial landing at the same multi-accessible points in the Julia set. Primitive majors are certain subsets of degree- @math -invariant laminations consisting of critical leaves and gaps. The space @math of primitive degree- @math majors is a spine for the set of monic degree- @math polynomials with distinct roots and serves as a parameterization of a subset of the boundary of the connectedness locus for degree- @math polynomials. The core entropy of a postcritically finite polynomial is the topological entropy of the action of the polynomial on the associated Hubbard tree. Core entropy may be computed directly, bypassing the Hubbard tree, using a combinatorial analogue of the Hubbard tree within the context of degree- @math -invariant laminations.
We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial f with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure μ this easily implies that one of the following holds: 1. for μ-a.e. x E J(f), ω(x) = J(f); 2. for μ-a.e. x ∈ J(f), ω(x) = ω(c(x)) for a critical point c(x) depending on x. Abstract We characterize the laminations associated to complex polynomials with connected Julia sets and without irrationally neutral cycles. Thurston defined invariant laminations, i.e. collections of chords of the unit circle S (called leaves) that are pairwise disjoint inside the open unit disk and satisfy a few dynamical properties. To be directly associated to a polynomial, a lamination has to be generated by an equivalence relation with specific properties on S; then it is called a q-lamination. Since not all laminations are q-laminations, then from the point of view of studying polynomials the most interesting are those of them which are limits of q-laminations. In this paper we introduce an alternative definition of an invariant lamination, which involves only conditions on the leaves (and avoids gap invariance). The new class of laminations is slightly smaller than that defined by Thurston and is closed. We use this notion to elucidate the connection between invariant laminations and invariant equivalence relations on S.  For a closed lamination on the unit circle invariant under z 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.