將此演算法應用於具備固定提升 \begin{align*} \Phi_c(x， y) = (x^d + cy^d， y^d)\quad \text{與}\quad \Psi(x， y) = (x^d， y^d) \end{align*} 的特定有理函數族時，我們根據次數 $d$ 的奇偶性與參數 $c$ 的正負號，識別並嚴格證明了截然不同的極值行為。對於偶數次數 $d$，我們證明了極值行為中存在著根本的分岔：當 $c \in \mathbb N$ 時，算術距離 $\hat{\Delta}(\Phi_c， \Psi)$ 隨 $c$ 嚴格遞增，並在常數座標 $(\pm 1， 1)$ 處達到全域上確界；然而，當 $c \in -\mathbb N$ 時，上確界轉移至對稱座標 $(\pm p， 1)$，其中 $p$ 對應於仿射映射 $\Phi_c(x， 1)$ 的唯一正實不動點。反之，對於奇數次數 $d$ 與參數 $c \in \mathbb Z \setminus \{0\}$，我們證明上確界必定在某點 $(x^*， 1) \in \mathbb A^2(\mathbb R)$ 處達到，其中 $x^*$ 恰好與該系統的一個實不動點重合。總結來說，這些發現建立了算術距離的全域最大值與底層動力系統的不變代數結構（特別是實週期點與前週期點）之間深刻的關聯。;This thesis investigates the analytic properties of Green functions and the extremal behavior of arithmetic distance in arithmetic dynamics. First， we study the modified Green function $\hat{\mathcal G}_{\Phi， v}$ associated with a rational map over a field equipped with an absolute value $v$. We establish its continuity properties with respect to the chordal metric $\rho_v$ by analyzing the Lipschitz constant $L_\phi$ of the associated morphism relative to its degree $d$. Specifically， when $L_\phi > d$， we prove that $\hat{\mathcal G}_{\Phi， v}$ is H\"older continuous with an explicit exponent $\frac{\log d}{\log L_\phi}$ and formulate the corresponding H\"older constant. Furthermore， we demonstrate that when $L_\phi < d$， the modified Green function is Lipschitz continuous， and we provide an explicit upper bound for the Lipschitz constant.
In the second part， the thesis explores the arithmetic distance $\hat{\delta}(\phi， \psi)$ between rational maps， defined as the supremum of the difference between their associated dynamical canonical heights $\hat{h}_\phi$ and $\hat{h}_\psi$. For rational maps with good reduction， we demonstrate that this distance can be reformulated into a computationally favorable form using modified Green functions. Specifically， the problem reduces to evaluating the supremum \begin{align*} \hat{\Delta}(\Phi， \Psi) = \sup_{[x:y] \in \mathbb P^1(\mathbb R)} \left| \hat{\mathcal G}_{\Phi， \infty}(x， y) - \hat{\mathcal G}_{\Psi， \infty}(x， y) \right|. \end{align*} To effectively compute this global supremum over the real projective line， we develop a novel algorithm: the Two-Pass Grid Search Algorithm with Dynamic Chart Switching. This hybrid approach utilizes exact theoretical exponents to filter suboptimal regions securely， and employs dynamic affine charts to prevent numerical overflow associated with unbounded affine coordinates.
Applying this algorithm to the specific family of rational functions with fixed lifts \begin{align*} \Phi_c(x， y) = (x^d + cy^d， y^d)\quad \text{and}\quad \Psi(x， y) = (x^d， y^d)， \end{align*} we identify and rigorously prove distinct extremal behaviors based on the parity of the degree $d$ and the sign of the parameter $c$. For even degrees $d$， we demonstrate a fundamental bifurcation in the extremal behavior: when $c \in \mathbb N$， the arithmetic distance $\hat{\Delta}(\Phi_c， \Psi)$ strictly increases with $c$ and globally attains its supremum at the constant coordinates $(\pm 1， 1)$; however， when $c \in -\mathbb N$， the supremum shifts to the symmetric coordinates $(\pm p， 1)$， where $p$ corresponds to the unique positive real fixed point of the affine map $\Phi_c(x， 1)$. Conversely， for odd degrees $d$ and parameters $c \in \mathbb Z \setminus \{0\}$， we demonstrate that the supremum is invariably attained at a point $(x^*， 1) \in \mathbb A^2(\mathbb R)$， where $x^*$ precisely coincides with a real fixed point of the system. Collectively， these findings establish a profound connection between the global maximum of the arithmetic distance and the invariant algebraic structures—specifically， the real periodic and pre-periodic points—of the underlying dynamical systems.
