[Paper Review] Central Strips of Sibling Leaves in Laminations of the Unit Disk

David J. Cosper, Jeffrey K. Houghton+3 more|arXiv (Cornell University)|Aug 1, 2014

Mathematical Dynamics and Fractals3 references5 citations

TL;DR

This paper generalizes Thurston's Central Strip Lemma from quadratic laminations to laminations of degree $d \geq 2$, introducing the concept of central strips for sibling leaves and proving a generalized Central Strip Lemma. The key contribution is a structural theorem that controls the dynamics of identity return polygons in higher-degree laminations, showing that such polygons cannot exist under certain geometric constraints, thereby extending results on wandering and periodic behavior in holomorphic dynamics.

ABSTRACT

Quadratic laminations of the unit disk were introduced by Thurston as a vehicle for understanding the (connected) Julia sets of quadratic polynomials and the parameter space of quadratic polynomials. The "Central Strip Lemma" plays a key role in Thurston's classification of gaps in quadratic laminations, and in describing the corresponding parameter space. We generalize the notion of {\em Central Strip} to laminations of all degrees $d\ge2$ and prove a Central Strip Lemma for degree $d\ge2$. We conclude with applications of the Central Strip Lemma to {\em identity return polygons} that show it may play a role similar to Thurston's lemma for higher degree laminations.

Motivation & Objective

• To extend Thurston's Central Strip Lemma from quadratic ($d=2$) laminations to laminations of arbitrary degree $d \geq 2$.
• To define and analyze the concept of 'central strips' in higher-degree laminations, particularly for sibling leaves.
• To establish conditions under which identity return polygons—periodic polygons returning to themselves under angle-doubling maps—cannot exist in $d$-invariant laminations.
• To provide a foundation for understanding the dynamics of higher-degree polynomial Julia sets and their parameter spaces through geometric control of leaf behavior near critical chords.

Proposed method

• Introduce the notion of a 'sibling portrait' to describe pairs of leaves sharing a common image under $\sigma_d$.
• Define the 'central strip' of a leaf as the region in the unit disk bounded by the leaf and its image under $\sigma_d$, with specific geometric and topological constraints.
• Prove the Central Strip Structure Theorem (Theorem 2.3), which characterizes the configuration of leaves and their images within central strips.
• Use the leaf length function $\tau_d(x)$ to analyze how chord lengths evolve under $\sigma_d$, identifying intervals where lengths increase or decrease.
• Apply the generalized Central Strip Lemma (Theorem 2.9) to show that if two long sides of a polygon approach critical chords, they must lie within a central strip, leading to contradiction if polygon is to return to itself.
• Use case analysis and geometric constraints (e.g., distance $\frac{1}{12}$ from critical chords) to rule out identity return quadrilaterals and triangles in $\sigma_3$.

Experimental results

Research questions

• RQ1 Can the Central Strip Lemma be generalized to laminations of degree $d \geq 3$, and what does it imply for the dynamics of periodic polygons?
• RQ2 Under what conditions can an identity return polygon exist in a $d$-invariant lamination for $d > 2$, and what geometric constraints must it satisfy?
• RQ3 Is there a bound on the number of identity return triangle orbits in a $3$-invariant lamination, and can such orbits be uniquely determined by their proximity to critical chords?
• RQ4 What is the minimal invariant lamination containing a given identity return triangle, and what structural properties must it satisfy?
• RQ5 How do the dynamics of $\sigma_d$ on $d$-gons relate to the structure of the parameter space of degree-$d$ polynomials?

Key findings

• The Central Strip Lemma is generalized to all $d \geq 2$, proving that if two long sides of a polygon approach the same critical chord, they must lie within a central strip, which restricts their configuration.
• No identity return quadrilateral exists in $\sigma_3$ laminations, as shown by contradiction using central strip constraints and distance bounds from critical chords.
• For $\sigma_3$, identity return triangles cannot exist if two sides simultaneously approach different critical chords within $\frac{1}{12}$ of their length, due to the central strip structure.
• The leaf length function $\tau_d(x)$ shows that leaves with length $< \frac{1}{d+1}$ grow under iteration until they reach or exceed critical thresholds, which is key to analyzing convergence to critical chords.
• The paper proves that in $\sigma_3$, a polygon cannot return to itself if its sides fail to approach critical chords within $\frac{1}{12}$, and such behavior leads to contradiction under the Central Strip Lemma.
• For $d=3$, a $3$-invariant lamination can contain at most one identity return triangle orbit where two sides approach different critical chords simultaneously within $\frac{1}{12}$ of their critical length.