Laminations in the language of leaves

TL;DR

This paper introduces a new, slightly smaller class of invariant laminations that simplifies Thurston's original definition, clarifying their connection to invariant equivalence relations on the circle.

Contribution

It proposes an alternative, leaf-only definition of invariant laminations, making the class more manageable and better understood in relation to polynomial dynamics.

Findings

The new class of laminations is closed and slightly smaller than Thurston's original class.

The paper clarifies the relationship between invariant laminations and invariant equivalence relations.

It provides a framework for studying limits of q-laminations in polynomial dynamics.

Abstract

Thurston defined invariant laminations, i.e. collections of chords of the unit circle $S^1$ (called \emph{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 $S^1$; then it is called a \emph{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 $S^1$.