A universal model for the bifurcations of asymptotic values

TL;DR

This paper introduces a universal model for bifurcations of asymptotic values in transcendental dynamics, defining a Tandelbrot set and proving a Straightening Theorem with applications to meromorphic map families.

Contribution

It develops a new framework for tangent-like maps, introduces the Tandelbrot set, and establishes a Straightening Theorem with implications for bifurcation analysis.

Findings

Defined tangent-like maps as transcendental analogues of polynomial-like maps.

Proved a Straightening Theorem for tangent-like maps with uniqueness conditions.

Demonstrated the existence of Tandelbrot set copies in bifurcation loci of meromorphic maps.

Abstract

We study the notion of tangent-like maps, which is a transcendental analogue of polynomial-like maps. We introduce a model family analogous to quadratic polynomials, with only one free asymptotic value, and define the "Tandelbrot set" as the analogue of the Mandelbrot set. We prove a Straightening Theorem for tangent-like maps, with uniqueness of the model map in the case where the filled-in Julia set is connected, and a parameter version of the Straightening Theorem for suitable holomorphic families of tangent-like maps. As a consequence, we prove the existence of topological copies of the Tandelbrot set in bifurcation loci of numerous families of meromorphic maps.