2022/06/11 by Tao Chen, Chen, Tao, Linda Keen +1
Mathematics · #2010 MSC Primary: 37F30 #30D30 #32A20 #37F10 #37F20 #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions #Secondary:30F30
paper · pdf · doi:10.48550/arxiv.2206.05622
openalex publication_date 2022/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, exp(z) + a, and the tangent family λtan z. The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values ± λi, is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between exp(z) + a and λtan z. Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be pole, the "polar asymptotic value" of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the exp(z) + a and λtan z families.