Algebraic billiards in the Fermat hyperbola

TL;DR

This paper investigates algebraic billiards in generic algebraic curves, showing quadratic growth of dynamical degree and measure-zero periodic points, and introduces a new billiard table, the Fermat hyperbola, with specialized stability properties.

Contribution

It establishes new results on the algebraic dynamics of billiards in algebraic curves and introduces the Fermat hyperbola as a key example with unique stability features.

Findings

Dynamical degree grows quadratically with degree d

Set of complex periodic points has measure zero

Constructs algebraically stable models for the billiard map

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.