Weak mixing of triangular billiards with an irrational angle

Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions. (Lean)

Manuscripts (2)

Weak mixing of triangular billiards with an irrational angle

October 5, 2026 · source · cite as \cite{OAI:Weak-mixing-of-triangular-billiards-with-an-irrational-angle-October-5-2026}

We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized Liouville measure. Equivalently, the product of the flow with itself is ergodic.

Ergodicity of triangular billiards with an irrational angle

September 25, 2026 · source · cite as \cite{OAI:Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026}

We prove that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is ergodic for normalized area times uniform angular measure. No genericity or Diophantine condition is required. The flow is considered outside the null set of trajectories that hit a vertex.

Result 150 · Dynamical systems and ergodic theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.