Algebraic billiards in the Fermat hyperbola

TL;DR

This work develops a rigorous algebraic-dynamics framework for billiards on algebraic plane curves, introducing a rational billiards correspondence $b_{C,D}$ on $C\times D$ formed from a secant map and a reflection. By specializing to the Fermat hyperbola, the authors exploit a favorable indeterminacy structure to construct birational models (notably $P$ and, for even $d$, $P_+$) in which the lifted map is essentially or algebraically stable, enabling precise dynamical-degree computations and a complete understanding of periodic orbits. They derive a quadratic growth bound for the dynamical degree in the degree $d$ and obtain an explicit formula for $\lambda_1(b)$ via a reduced action on a small divisor group, as well as a detailed description of indeterminacy and exceptional sets. These results yield the Ivrii conjecture for generic complex algebraic billiards and extend to real domains with algebraically independent coefficients, thereby linking algebraic dynamics, birational geometry, and spectral theory of billiards. Overall, the paper provides a robust algebraic approach to chaotic behavior in algebraic billiards and demonstrates powerful stability phenomena through iterative blowups.

Abstract

We prove two results on the algebraic dynamics of billiards in generic algebraic curves of degree $d \geq 2$. First, the dynamical degree grows quadratically in $d$; second, the set of complex periodic points has measure 0, implying the Ivrii Conjecture for the classical billiard map in generic algebraic domains. To prove these results, we specialize to a new billiard table, the Fermat hyperbola, on which the indeterminacy points satisfy an exceptionality property. Over $\mathbb{C}$, we construct an algebraically stable model for this billiard via an iterated blowup. Over more general fields, we prove essential stability, i.e. algebraic stability for a particular big and nef divisor.

Figures (6)

• Figure 1: Algebraic billiards in the curve $C$ defined by $x_1 = x_0^3$ relative to the quadratic form $\Theta = q_0^2 + q_1^2$. A generic input (represented by the leftmost vector) has two images (represented by the middle and rightmost vectors).
• Figure 2: The real locus of the Fermat hyperbola of degree $d=2,3,4,5$.
• Figure 3: Model $P$, $d = 4$.
• Figure 4: Dynamics on $\mathcal{E}^{(p)}$, $p \in \mathop{\mathrm{Ind}}\nolimits r$, when $d = 5$. Read the figure as a chain of five Riemann spheres viewed in profile. The top sphere is $\hat{D}^{(p)} = E_0^{(p)}$. The bottom four spheres form $\mathcal{E}^{(p)}$. The reflection map $\hat{r}$ rotates the diagram, exchanging the top and bottom sphere. The horizontal line depicts the unit circle on $E^{(p)}_{\mathop{\mathrm{mid}}\nolimits}$. The two dots above the unit circle are $\mathop{\mathrm{Exc}}\nolimits \hat{b} \cap \mathcal{E}^{(p)}$; the only contracted curve is $E_1^{(p)}$. The two dots below the unit circle are $\mathop{\mathrm{Ind}}\nolimits \hat{b} \cap \mathcal{E}^{(p)}$. The map $\hat{b}$ sends the upper half of $E^{(p)}_{\mathop{\mathrm{mid}}\nolimits}$ into itself, so the forward orbit of $\mathop{\mathrm{Exc}}\nolimits \hat{b}$ never meets $\mathop{\mathrm{Ind}}\nolimits \hat{b}$.
• Figure 5: Model $P_+$, $d = 4$. The dots in $P$ represent blowup centers.

Theorems & Definitions (113)

• Definition 1.1: MR4048444
• Theorem 1.2
• Conjecture 1.3: billiardsI
• Conjecture 1.4: Ivrii Conjecture
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3