A Partial Characterization of Cosine Thurston Maps

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 $C_{\lambda }(z)=\lambda \cos z$ for a unique $\lambda \in {\mathbb{C}}^{\ast }$ if only if it has no degenerate Levy cycle.