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We introduce an algebraic formulation of billiards on plane curves overalgebraically closed fields, extending Glutsyuk's complex billiards. For anysmooth algebraic curve $C$ of degree $d \geq 2$, algebraic billiards is arational $(d-1)$-to-$(d-1)$ surface correspondence on the space of unitcotangent vectors based on $C$. We prove that the dynamical degree of thebilliards 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, forsmooth real algebraic curves, the topological entropy of the classicalbilliards map is at most $\log \rho_d$. We further show that the billiardscorrespondence satisfies the singularity confinement property and preserves anatural $2$-form. To prove our bounds, we construct a birational model thatpartially resolves the indeterminacy of algebraic billiards.

The dynamical degree of billiards in an algebraic curve