The dynamical degree of billiards in an algebraic curve

TL;DR

This work develops an algebraic framework for billiards on plane curves over algebraically closed fields by defining a billiards correspondence b on C × D, where C has degree d ≥ 2. The authors prove a uniform upper bound on the first dynamical degree, λ1(b) ≤ ρ_d, with ρ_d the largest root of a cubic Φ_d, yielding explicit bounds λ1(b) < 2d^2 - d - 3 and, for real curves, h_top ≤ log ρ_d. Central to the approach is a decomposition b = r ∘ s into a secant map s and a reflection map r, the construction of a modified phase space P obtained by blowing up scratch points to resolve indeterminacies, and the analysis of b_* on Num^1(P); singularity confinement further clarifies orbit behavior and supports the stability arguments. The paper also establishes an invariant rational 2-form ω on C × D, studies the conic case d = 2 (ellipses) leading to complete integrability and a new proof of Poncelet porism, and discusses real-billiard consequences, including entropy bounds and orbit-growth rates. Overall, the work extends Glutsyuk’s complex billiards to an algebraic setting, connects dynamical degrees with geometric features of the table, and provides tools to bound entropy and understand orbit complexity in real and complex billiards.

Abstract

We introduce an algebraic formulation of billiards on plane curves over algebraically closed fields, extending Glutsyuk's complex billiards. For any smooth algebraic curve $C$ of degree $d \geq 2$, algebraic billiards is a rational $(d-1)$-to-$(d-1)$ surface correspondence on the space of unit tangent vectors based on $C$. We prove that the dynamical degree of the billiards correspondence is at most an explicit cubic algebraic integer $ρ_d < 2d^2 - d - 3$, depending only on the degree $d$ of $C$. As a corollary, for smooth real algebraic curves, the topological entropy of the classical billiards map is at most $\log ρ_d$. We further show that the billiards correspondence satisfies the singularity confinement property and preserves a natural $2$-form. To prove our bounds, we construct a birational model that partially resolves the indeterminacy of algebraic billiards.

Figures (4)

• Figure 1: Over algebraically closed fields such as $\mathbb C$, the billiards map is multivalued. Since non-real intersections of the ray with $C$ cannot be drawn, we show here a real point $c$ of $C$ and a real direction $q$ such that the ray has only real intersection points with $C$. Note that the notion of "first point of intersection" of a ray with $C$ does not extend to general $(c, q)$.
• Figure 2: The secant correspondence $s$ on the real locus of a plane cubic $C$. A point in the domain is represented as a unit-length vector based on $C$, given by a base point $c \in C$ and direction $q \in S^1$. The two images of $(c, q)$ are the points $(c', q)$. Copies of $S^1$ are shown based at each $c'$ to show that the value of $q$ does not change upon applying $s$.
• Figure 3: The reflection correspondence $r$ on the real locus of a plane curve $C$. The point $(c, q)$ is mapped to $(c, q')$. The point $q'$ is the point on $D$ beside $q$ on the line through $q$ parallel to the normal $n_c(C)$ through $c$ at $C$.
• Figure 4: Singularity confinement for billiards. Here we show a hyperbola $C$ with a horizontal asymptote, corresponding to a point $c_\infty = [1 : 0 : 0]$. Let $c \in C$. The three dotted lines are the asymptote, the horizontal line $\ell$ through $c$, and the reflection of $\ell$ across the asymptote. As the direction of the first shot tends to $[1 : 0 : 1]$, the limiting position of $c"$ is the intersection of the lower dotted line with $C$. Thus the limit of $c"$ depends on $c$, but the limit of $c'$ does not.

Theorems & Definitions (101)

• Definition 1.1
• Definition 1.2
• Definition 1.3: MR4048444
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 1.7
• Remark 1.8
• Corollary 1.10
• Corollary 1.11