A Partial Characterization of Cosine Thurston Maps

TL;DR

This work extends Thurston’s topological characterization to the cosine family by constructing postsingularly finite topological cosine maps and formulating a cosine analogue of the Thurston pullback theory on a suitable Teichmüller space. It introduces cos‑efficient quadratic differentials and a mass condition to control how mass distributes under the cosine push-forward, enabling contraction arguments and a fixed-point criterion. The main result provides a partial exponential-like characterization: a postsingularly finite cosine map with strictly preperiodic critical points is combinatorially equivalent to a unique holomorphic cosine map $C_\lambda$ if and only if no degenerate Levy cycle arises, under the mass condition. The paper also sketches pathways to generalize these ideas to broader transcendental maps, highlighting the role of critical points and the potential to broaden the scope of transcendental Thurston theory.

Abstract

In this paper, we introduce cosine Thurston maps. In particular, we construct postsingularly finite topological cosine maps and focus on such maps with strictly preperiodic critical points. We use the techniques of Hubbard, Schleicher, and Shishikura to prove that, subject to a condition on the critical points, a postsingularly finite topological cosine map with strictly preperiodic critical points is combinatorially equivalent to $C_λ(z) = λ\cos z$ for a unique $λ\in \mathbb{C}^*$ if only if it has no degenerate Levy cycle.

Figures (6)

• Figure 1: A topological cosine map $f:S^2 \setminus \{\infty\} \rightarrow S^2 \setminus \{\infty\}$.
• Figure 2: Initial step in the construction of the Thurston pullback map for $C_\lambda(z) = \lambda \cos z$.
• Figure 3: A candidate cos-efficient sequence of quadratic differentials is pictured here. Each quadratic differential has 6 poles and 2 zeroes, and mass $4$ with the majority of its mass concentrated in a small neighbourhood of $1$. The polygonal representation shows that an annulus of very large modulus separates $1$ from $0$ and $\infty$, which then gives an interpretation for how close poles are in $\mathbb{C}$. The line connecting $0$ to $\infty$ in the polygonal representation becomes the negative real axis in $\mathbb{C}$, as shown.
• Figure 4: Conformal mapping of Figure \ref{['fig:quad_diff']} with various radii labeled
• Figure 5: A candidate cos-efficient sequence of quadratic differentials is pictured here. Each quadratic differential has 6 poles and 2 zeroes, and mass $4$ with the majority of its mass concentrated in a small neighbourhood of $0$. The interpretation of the polygonal and plane representation is similar to Figure \ref{['fig:quad_diff']}.

Theorems & Definitions (40)

• Theorem 1.1: Partial characterization of cosine maps - preperiodic case
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Example 2.7
• Remark 2.8
• Theorem 2.9: Relationship between cosine and fixed point of $\sigma_f$