Abstract
AbstractIn this paper, we investigate the precise behavior of orbits inside attracting basins. Let f be a holomorphic polynomial of degree $$m\ge 2$$
m
≥
2
in $$\mathbb {C}$$
C
, $$\mathcal {A}(p)$$
A
(
p
)
be the basin of attraction of an attracting fixed point p of f, and $$\Omega _i (i=1, 2, \cdots )$$
Ω
i
(
i
=
1
,
2
,
⋯
)
be the connected components of $$\mathcal {A}(p)$$
A
(
p
)
. Assume $$\Omega _1$$
Ω
1
contains p and $$\{f^{-1}(p)\}\cap \Omega _1\ne \{p\}$$
{
f
-
1
(
p
)
}
∩
Ω
1
≠
{
p
}
. Then there is a constant C so that for every point $$z_0$$
z
0
inside any $$\Omega _i$$
Ω
i
, there exists a point $$q\in \cup _k f^{-k}(p)$$
q
∈
∪
k
f
-
k
(
p
)
inside $$\Omega _i$$
Ω
i
such that $$d_{\Omega _i}(z_0, q)\le C$$
d
Ω
i
(
z
0
,
q
)
≤
C
, where $$d_{\Omega _i}$$
d
Ω
i
is the Kobayashi distance on $$\Omega _i.$$
Ω
i
.
In paper Hu (Dynamics inside parabolic basins, 2022), we proved that this result is not valid for parabolic basins.
Funder
Università degli Studi di Parma
Publisher
Springer Science and Business Media LLC