Algebraic billiards in the Fermat hyperbola

Algebraic billiards in a plane curve of degree d ≥ 2 is a rational correspondence on a surface. The dynamical degree is an algebraic analogue of entropy that measures intersection-theoretic complexity. We compute a lower bound on the dynamical degree of billiards in the generic algebraic curve of degree d over any field of characteristic coprime to $2d$, complementing the upper bound in our previous work. To do this, we specialize to a highly symmetric curve that we call the Fermat hyperbola. Over C, we construct an algebraically stable model for this billiard via an iterated blowup. Over general fields, we compute the growth of a particular big and nef divisor.