Explicit Families and Distribution of Triangular Billiards of Weakly Exponential Growth

Abstract

Recently, it has been shown that the combinatorial complexity function $N_c(n)$ of a typical triangular billiard has weakly exponential growth, i.e., for almost any triangle and any $\varepsilon>0$ there is a constant $C$ such that $N_c(n)<Ce^{n^\varepsilon}$. We give the first example, an infinite family of explicitly described triangles with weakly exponential complexity growth. The simplest example in this family is a right triangle with integer side lengths. Moreover, we find one-parameter families of triangular tables with weakly exponential growth for generic parameters.