Dynamics inside Parabolic Basins

Mi Hu

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arXiv · 2208.03756

Dynamics inside Parabolic Basins

Mi Hu

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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.

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Mi Hu. 2022-08-07. Dynamics inside Parabolic Basins. https://arxiv.org/abs/2208.03756

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