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.
Not all degree @math -invariant laminations correspond to degree @math polynomials. One necessary condition for a lamination to be directly associated to a polynomial is that it be generated by a tree-like equivalence relation on @math (see, for example, @cite_20 @cite_28 @cite_23 @cite_13 and Section 2 of Part I). To distinguish between arbitrary invariant laminations and those associated to polynomials, we call the invariant laminations defined by Thurston geometric invariant laminations , and the smaller class of laminations defined by tree-like equivalence relations combinatorial invariant laminations . (In @cite_16 , such laminations are called @math -laminations).
@cite_28 @cite_23 @cite_16 @cite_13 @cite_20
1978487894 2150678455 2065583259 1971043610
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. Abstract We characterize the laminations associated to complex polynomials with connected Julia sets and without irrationally neutral cycles. This paper presents a method for automatically generating all polynomial invariants in simple loops. It is first shown that the set of polynomials serving as loop invariants has the algebraic structure of an ideal. Based on this connection, a fixpoint procedure using operations on ideals and Grobner basis constructions is proposed for finding all polynomial invariants. Most importantly, it is proved that the procedure terminates in at most m+1 iterations, where m is the number of program variables. The proof relies on showing that the irreducible components of the varieties associated with the ideals generated by the procedure either remain the same or increase their dimension at every iteration of the fixpoint procedure. This yields a correct and complete algorithm for inferring conjunctions of polynomial equalities as invariants. The method has been implemented in Maple using the Groebner package. The implementation has been used to automatically discover non-trivial invariants for several examples to illustrate the power of the technique.