Explicit Arithmetic Distance Motivated by Tits Alternative in Arithmetic Dynamics
Wayne Peng (joint with You-Ting Chen)
National Central University
<h4>Previous Result</h4> <div class="theorem"> <div class="theorem-header"> <span class="theorem-authors">(Bell-Huang-P.-Tom, JEMS, 2020)</span> </div> <p> Let $f$ and $g$ be rational functions of degrees greater than $2$. Let $\hat{h}_f$ and $\hat{h}_g$ be the canonical height of $f$ and $g$ respectively. We have $\hat{h}_f\neq \hat{h}_g$ if and only if there exists an integer $j$ such that $\langle f^j,g^j\rangle$ is a rank 2 free generated semigroup. </p> </div> <div class="proof no-auto-qed fragment"> <p> Let $\omega=\phi_m\cdots\phi_1$ where $\phi_j$ is equal to $f$ or $g$ for each $i$. Let $s_j=\phi_j\cdots \phi_1$ for $j\leq m$.</p> <p class="fragment"> Let $h$ be the Weil height. Let $C_f$ be the constant such that $$|h(f(a))-dh(a)|< C_f.$$ </p> <p class="fragment"> Then, we have $$ \left|\dfrac{h(w(\alpha))}{\deg\omega}-\dfrac{h(s_j(\alpha))}{\deg s_j}\right|<\dfrac{\max\{C_f,C_g\}}{\min\{\deg(f),\deg(g)\}^j} $$ for $j=1,\ldots,m$. </p> </div>
<div class="proof unlabelled"> <p class="fragment">We choose an $\varepsilon< |h_f(\alpha)-h_g(\alpha)|/2\neq 0$. Then $B(h_f,\varepsilon)\cap B(h_g,\varepsilon)=\emptyset$.</p><p class="fragment"> By the inequality in the previous slides, we know there is a $j$ such that, for all words starting with $j$ many $f$, their canonical heights at $\alpha$ will stay in $B(h_f,\varepsilon)$.</p><p class="fragment"> Similarly, for all words starting with $j$ many $g$, their canonical heights at $\alpha$ will stay in $B(h_g,\varepsilon)$.</p><p class="fragment"> Thus we can conclude that any word starting with $j$ many $f$ won't equal to one starting with $j$ many $g$.</p> </div>
<div class="lemma"> <div class="lemma-header"></div> <p> Fix a constant $B>0$ and an integer $d>1$. For any number field $K$, we have $C_f>B$ for all but finitely many degree $d>1$ rational maps $f:\mathbb{P}^1(K)\to\mathbb{P}^1(K)$. </p> </div> <div class="proof fragment"> <p> <ul> <li class="fragment">A raional map of degree $d$ is uniquely determined by $2d+1$ points.</li> <li class="fragment">Let $S\subseteq\mathbb P^1(K)$ be a finite set with $|S|\geq 2d+1$.</li> <li class="fragment">$C_f$ gives the bound of the image $f(S)$ by height.</li> <li class="fragment">Apply Northcott.</li> </ul> </p> </div>
<div class="theorem"> <div class="theorem-header"> <span class="theorem-authors">(P., unpublished)</span> </div> <p> If one of the degrees of $f$ or $g$ is strickly greater than $3$, then the semigroup $\langle f,g\rangle$ is free for all but finitely many $f$ and $g$. In particular, the semigroup $\langle f^2, g\rangle$ is free for all but finitely many $f$. </p> </div> <div class="proof no-auto-qed"> <p class="fragment"> We have $$ \dfrac{C_f}{d}\left(1-\dfrac{1}{d-1}\right)<|h_f(a)-h(a)|<\dfrac{C_f}{d-1} $$ and $$ \left|h_f(P)-\dfrac{h(f^i(P))}{d^i}\right|<\dfrac{C_f}{d^i(d-1)} $$ for $a\in\mathbb{P}^1(\bar{Q})$. </p> <p class="fragment"> It follows that $$ \begin{align*} |\hat{h}_f(P)-\hat{h}_g(P)|&\geq |\hat{h}_f(P)-h(P)|-|h(P)-\hat{h}_g(P)|\\ &\geq \dfrac{C_f}{d_f}\left(1-\dfrac{1}{d_f-1}\right)-\frac{C_g}{d_g-1}. \end{align*} $$ </p> </div>
<div class="proof unlabelled"> <p> By the lemma, we have $$ C_f>\frac{d_f(d_f-1)(d_g+1)}{(d_g-3)d_g(d_g-1)}C_g $$ for all but finitely many $g$ with $d_g>3$. </p> <p class="fragment"> Putting these inequality together, we have $$ \begin{align*} \dfrac{C_f}{d_f}\left(1-\dfrac{1}{d_f-1}\right)-&\frac{C_g}{d_g-1} = \frac{(d_f-2)d_g(d_g-1)C_f-d_f(d_f-1)d_g C_g}{d_f(d_f-1)d_g(d_g-1)}\\ =& \frac{d_g(d_g-1)C_f+d_f(d_f-1)C_g+d_g(d_g-1)(d_f-3)C_f-d_f(d_f-1)(d_g+1)C_g}{d_f(d_f-1)d_g(d_g-1)}\\ >& \frac{C_f}{d_f(d_f-1)}+\frac{C_g}{d_g(d_g-1)}. \end{align*} $$ </p> </div>
<h4>Continuity of dynamical canonical heights</h4> <div class="definition"> <p> Let $f,g:\mathbb{P}^1(\mathbb{Q})\to \mathbb{P}^1(\mathbb{Q})$. The <em>arithmetic distance</em> between $f$ and $g$ is defined by $$ \hat\Delta(f,g)=\sup_{x\in\mathbb{P}^1(\mathbb{Q})}|\hat{h}_f(x)-\hat{h}_g(x)|. $$ </p> </div> <p class="fragment"> <b>Goal</b>: find $\hat\Delta(f,g)$. </p> <div class="definition fragment"> <p> Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces, and $f: X \to Y$ be a function. The function $f$ is said to be <em>Hölder continuous</em> if $$ \exists\ C, \alpha > 0: \forall\ x, y \in X: d_Y(f(x), f(y)) \le C d_X(x,y)^\alpha. $$ When $\alpha=1$ is allowed, it is said to be <em>Lipschitz continuous</em>. </p> </div>
<h4>A General Strategy</h4> <p class="fragment"> Let $\phi:I\to\mathbb{R}_{\geq 0}$ be Hölder continuous with Hölder constant $C$ and Hölder exponent $\alpha$.</p><p class="fragment"> Given a partition $\mathcal P=\{x_0,x_1,\ldots, x_N\}\subset I$, let $\|\mathcal P\|=\max\{x_i-x_{i-1}|i=0,\ldots, N-1\}$.</p><p class="fragment"> We then have $$\forall x\in (x_{i-1},x_{i+1}): |\phi(x)-\phi(x_i)|\leq C|x-x_i|^\alpha<\|\mathcal P\|^\alpha. $$ </p> <p class="fragment"> It implies $$\phi(x)<\phi(x_i)+C\|\mathcal P\|^\alpha.$$ Thus, we may discard all $x_i$ if $$|\phi(x_i)|<\max\{\phi(x)|x\in\mathcal {P}\}-C|\mathcal P|^\alpha.$$ </p>
### Algorithm: Hölder Maximum Search **Input:** - $\phi$ : Hölder continuous function - $I$ : initial interval - $C$ : Hölder constant - $\alpha$ : Hölder exponent - $\varepsilon$ : stopping tolerance - $N$ : number of subdivisions per interval **Output:** - Midpoints of candidate intervals containing the global maximum
<pre><code> 1. ActiveIntervals ← {I} 2. while ∃ J ∈ ActiveIntervals with |J| > ε do 3. NewIntervals ← ∅ 4. for each interval J=[a,b] ∈ ActiveIntervals do 5. Partition J into P={x₀,x₁,...,xₙ} 6. h ← max{xᵢ−xᵢ₋₁ : i=1,...,N} 7. M ← max{φ(xᵢ) : xᵢ∈P} 8. for i=1,...,N−1 do 9. if φ(xᵢ) ≥ M − Chᵅ then 10. NewIntervals ← NewIntervals ∪ {[xᵢ₋₁,xᵢ₊₁]} 11. end if 12. end for 13. end for 14. Merge overlapping intervals 15. ActiveIntervals ← NewIntervals 16. end while 17. Result ← ∅ 18. for each interval [a,b] ∈ ActiveIntervals do 19. Result ← Result ∪ {(a+b)/2} 20. end for 21. return Result </code></pre>
<h4>Notations</h4> <ul> <li class="fragment">$\Delta(f,g)(x)=|h_f(x)-h_g(x)|$</li> <li class="fragment">$K$ a number field (usually just $\mathbb{Q}$)</li> <li class="fragment">$\phi,\ \psi:\mathbb{P}^1(K)\to\mathbb{P}^1(K)$</li> <li class="fragment">$\Phi,\ \Psi:\mathbb{A}^2(K)\to\mathbb{P}^2(K)$ be lifts of $\phi$ and $\psi$ respectively; a lift of $\phi$ is a morphism such that $\Phi\circ \pi = \pi\circ \phi$ where $\pi:\mathbb{A}^2\to\mathbb{P}^1$.</li> </ul>
<h4>Good Reduction</h4> <div class="definition"> <p> A rational map $\phi:\mathbb P^1(K) \to \mathbb P^1(K)$ is said to have <em>good reduction modulo $\mathfrak p$</em> if it satisfies $\deg\phi = \deg\widetilde{\phi}$ where $\widetilde{\phi}$ is the reduced rational map defined over $\mathfrak O_K/\mathfrak p$. </p> </div> <div class="theorem fragment"> <div class="theorem-header"> </div> <p> Let $K$ be a field with an absolute value $v$ and $\phi : \mathbb P^1 \to \mathbb P^1$ be a morphism of degree $d \ge 2$. Let $\Phi = (F,G) : \mathbb A^2 \to \mathbb A^2$ be a lift of $\phi$. If $v$ is nonarchimedean and the map $\phi = [F,G] : \mathbb P^1 \to \mathbb P^1$ has good reduction at $v$, then $$\forall (x,y) \in \mathbb A_*^2(K): \mathcal G_{\Phi, v}(x, y) = \log \max(|x|_v, |y|_v).$$ </p> </div>
<div class="theorem"> <div class="theorem-header"> <span class="theorem-authors">Briend-Duval</span> </div> <p> For an endomorphism $\phi$ on $\mathbb P^N$ over $\mathbb C$ of degeree $d$, the field of complex numbers, take a lift $\Phi = (\Phi_0, ..., \Phi_N) : \mathbb C^{N+1} \to \mathbb C^{N+1}$ of $\phi$, and denote by $G$ the Green function of $\Phi$. Put $L_\phi = \sup_{G(z) \leq 0} \max |\Phi'_0(z), ..., \Phi'_N(z)|$. Then for any $\alpha < (\log d)/(\log L_\phi)$, $G$ is Hölder continuous with Hölder exponent $\alpha$. </p> </div>
<h3>Arithmetic Distance Bounding Flowchart</h3> <ul> <li class="fragment">$|\hat{h}_\phi-\hat{h}_\psi|$</li> <li class="fragment">$|G_{\Phi,v}-G_{\Psi,v}|$</li> <li class="fragment">$|G_{\Phi,v}(P)-G_{\Psi,v}(Q)|$</li> <li class="fragment">$|\hat{G}_{\Phi,v}(P)-\hat{G}_{\Psi,v}(Q)|+|\log\|P\|_v-\log\|Q\|_v|$</li> <li class="fragment">$\sum_{n=1}^\infty \dfrac{g(\phi^n(p))}{d^n}-\sum_{n=1}^\infty\dfrac{g(\phi^n(q))}{d^n}$</li> <li class="fragment">Using the fact that $\phi$ and $g$ are Lipschtiz, we find hölder constant and exponent. In particular, $L_\phi$ is the Lipschtiz constant of $\phi$</li> </ul>
<div class="lemma"> <div class="lemma-header"> <span class="theorem-authors">Chen-P.</span> </div> <p> The modified Green function $\hat{\mathcal G}_{\Phi, v}$ is Hölder continuous with respect to the chordal metric $\rho_v$ on $\mathbb P^1(\mathbb{Q})$. More precisely, $$\forall p,q\in\mathbb{P}^1(K):\left|\hat{\mathcal G}_{\Phi,v}(p)-\hat{\mathcal G}_{\Phi,v}(q)\right|\leq C_{g,\phi}\rho_v(P,Q)^{\frac{\log d}{\log L_\phi}}$$ where $L_\phi=\frac{2M_\phi}{c_1^2}$ is a Lipschitz constant of $\phi$; $L_g$ is a Lipschitz constant of $g$; $M_\phi=|H|_v$ with $$\Phi_0(x_0,y_0)\Phi_1(x_1,y_1)-\Phi_0(x_1,y_1)\Phi_1(x_0,y_0)=H\cdot (x_0y_1-x_1y_0),$$ and $c_1\leq \frac{\|\Phi(x,y)\|_v}{\|(x,y)\|_v}\leq c_2$. </p> </div>
<div class="proof"> <p> Let $p=\pi((x_0,y_0))$ and $q=\pi((x_1,y_1))\in\mathbb P^1$. We then have $$ \begin{align*} \rho_v(\phi(p),\phi(q))&=\frac{\Phi(x_0,y_0)\Phi(x_1,y_1)-\Phi(x_1,y_1)\Phi(x_0,y_0)}{\|\Phi(x_0,y_0)\|_v\|\Phi(x_1,y_1)\|_v}\\ &\leq \dfrac{\|(x_0,y_0)\|_v\|(x_1,y_1)\|_v}{\|\Phi(x_0,y_0)\|_v\|\Phi(x_1,y_1)\|_v}\dfrac{|H(x_0,y_0,x_1,y_1)|_v|x_0y_1-x_1y_0|_v}{\|(x_0,y_0)\|_v\|(x_1,y_1)\|_v}\\ &\leq \frac{1}{c^2}M_\phi\frac{|x_0y_1-x_1y_0|_v}{\|(x_0,y_0)\|_v\|(x_1,y_1)\|_v}\leq \frac{2}{c^2}M_\phi\rho(p,q) \end{align*} $$ </p> </div>
#### Application <div class="theorem"> <div class="theorem-header"> <span class="theorem-authors">Chen-P.</span> </div> <p> Let $d \in \mathbb N$ be even and $$ \phi_c(x,y) = \left[ x^d + cy^d:y^d \right]\text{ and }\psi(x,y) = \left[ x^d:y^d \right], $$ be rational functions with positive $c\in\mathbb{Z}$. Then we have $$ \hat\Delta(\Phi_c, \Psi) = \Delta_{\Phi_c, \Psi}(\pm 1,1), $$ and this quantity is increasing in $c$. </p> </div>
<div class="proof"> <p> We have $$\hat\delta(\Phi_c,\Psi)=\sup_{x\in\mathbb{Q}}\left|\lim_{n\to\infty}\frac{\log|F_c^n(x)|_\infty}{d^n}-\log\max\{|x|,1\}\right|$$ where $F_c(x)=x^d+c$. Let $x=\frac{a}{b}$. It follows that if $|x|<1$, then $$F_c^n\left(\frac{a}{b}\right)=\sum_{i=0}^{d^n}k_i\frac{a^i b^{d^n-i}}{b^{d^n}};$$ if $|x|>1$, then $$\frac{|F^n_c\frac{a}{b}|}{|x|^{d^n}}=\sum_{i=0}^{d^n}k_i\left(\frac{a^ib^{d^n-i}}{a^{d^n}}\right).$$ Both are smaller than $\Delta_{\Phi_c,\Psi}(1,1)$ since $k_i$ are all positive due to positive $c$. </p> </div>
<div class="theorem"> <div class="theorem-header"> <span class="theorem-authors">Chen-P.</span> </div> <p> Let $d$ be even and $c$ be negative, or let $d$ be odd. we have $$ \hat \Delta(\Phi_c, \Psi) = \log |r| $$ where $r$ is the fixed point of $f_c(x)=x^d+c$. </p> </div>
<div class="proof"> <p> First, we have $\hat{\Delta}(\Phi_c,\Psi)=\hat\Delta(\Phi_{-c},\Psi)$ since $\Phi_c$ is conjugate to $\Phi_{-c}$ by $-x$. Thus, we may assume that $c>0$. Using the same argument like the previous proof, we can show that $\Delta(\Phi_c,\Psi)(1,1)\geq \Delta(\Phi_c,\Psi)(x,1)$ for $x\geq 0$. </p> <p class="fragment"> Hence, we remian to show the following: </p> <ol> <li class="fragment">$\Delta(\Phi_c,\Psi)(1,1)\leq \log |r|$</li> <li class="fragment">$\Delta(\Phi_c,\Psi)(x,1)\leq \log |r|$ for $x\in(-|r|,0)$</li> <li class="fragment">$\Delta(\Phi_c,\Psi)(x,1)\leq \log |r|$ for $x\in(-\infty,-|r|)$</li> </ol> <p class="fragment"> All cases are more or less using the induction to show that $f_c^n(x)$ is bounded by $r^{d^n}$. </p> </div>
#### complexity on moduli space We define $$\mu([f])=\inf_{l\in\operatorname{PGL}_2(\bar{\mathbb{Q}})}\hat\Delta(f^l,x^d).$$ Stratagy: - Move the fixed points to $0$ and $\infty$. Reduce the freedom of mobius transformation from three to one. - Use our algorithm to optimize the last freedom choice.
Thank you