The dynamical degree of billiards in an algebraic curve

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 $\rho_d < 2d^2 - d - 3$, 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 $\log \rho_d$. 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.