Complete left tail asymptotic for the density of branching processes in the Schröder case

Abstract

For the density of Galton-Watson processes in the Schröder case, we derive a complete left tail asymptotic series consisting of power terms multiplied by periodic factors.

Figures (3)

• Figure 1: Gray area is the domain of definition of $K(z)$. Paths $\gamma$ and $\log_E\gamma$ are plotted with the blue color.
• Figure 2: Two examples of Julia sets computed with the help of https://www.marksmath.org/visualization/polynomial_julia_sets/$^1$: the cases $p_1=0.1$, $p_2=0.5$, $p_3=0.4$, and $p_1=0.1$, $p_2=0.1$, $p_3=0.5$, $p_4=0.3$.
• Figure 3: Two examples: the cases $p_1=0.1$, $p_2=0.5$, $p_3=0.4$ (upper curves), and $p_1=0.1$, $p_2=0.1$, $p_3=0.5$, $p_4=0.3$.