Dynamics inside Parabolic Basins

TL;DR

This paper studies the dynamics of orbits within parabolic basins of complex functions, demonstrating that certain points can be chosen so that their orbits stay a fixed Kobayashi distance away from a reference point, unlike in attracting basins.

Contribution

It establishes a new property of orbits in parabolic basins related to Kobayashi distances, contrasting with previous results on attracting basins.

Findings

Existence of points with orbits maintaining a fixed Kobayashi distance

Contrast with behavior in attracting basins

Extension of dynamical properties in complex analysis

Abstract

In this paper, we investigate the behavior of orbits inside parabolic basins. Let $f(z)=z+az^{m+1}+(\text{higher terms}), m\geq1, a\neq0.$ We choose an arbitrary constant $C>0$ and a point $q\in{\bf v_j}\cap\mathcal{P}_j$. Then there exists a point $z_0\in \mathcal{P}_j$ so that for any $\tilde{q}\in Q:= \cup_{l=0}^{\infty}f^{-l}(f^k(q)) (l, k$ are non-negative integers), the Kobayashi distance $d_{\mathcal {P}_j}(z_0, \tilde{q})> C$, where $d_{\mathcal{P}_j}$ is the Kobayashi metric. In a previous paper [4], we showed that this result is not valid for attracting basins.