Complex dynamics inside fatou sets

This thesis investigates the behavior of orbits inside Fatou sets in one dimension and higher dimensions. Suppose $f(z)$ is a polynomial of degree $N\geq 2$ on $\mathbb{C}$, $p$ is an attracting fixed point of $f(z),$ $\Omega_1$ is the immediate basin of attraction of $p$, $\{f^{-1}(p)\}\cap \Omega_1\neq\{p\}$, $\mathcal{A}(p)$ is the basin of attraction of $p$, $\Omega_i (i=1, 2, \cdots)$ are the connected components of $\mathcal{A}(p)$. Then there is a constant $C$ so that for every point $z_0$ inside any $\Omega_i$, there exists a point $q\in \cup_k \{f^{-k}(p)\}$ inside $\Omega_i$ such that $d_{\Omega_i}(z_0, q)\leq C$, where $d_{\Omega_i}$ is the Kobayashi distance on $\Omega_i.$ This result implies that the orbit of $z_0$ is tracked by the orbit of some point $q\in\cup_k\{f^{-k}(p)\}$. In addition, we prove a variety of different results related to the same questions about attracting basins for rational functions on $\hat{\mathbb{C}}$ and entire functions on $\mathbb{C},$ parabolic basins of holomorphic polynomials on $\mathbb{C}$, and finally, focus on attracting basins for some holomorphic polynomial maps on $\mathbb{C}^2.$