Ergodicity of geodesic flows on Aravinda-Farrell twisted doubles

Abstract

The announced rank-one ergodicity theorem of Rodriguez Hertz, Rodriguez Hertz, and Wu includes the nonpositively curved metrics considered here on Aravinda--Farrell twisted doubles. For these construction metrics, a direct geometric argument identifies the singular set. Their singular geodesics are precisely those tangent to the torus sections of finitely many flat product collars, so their initial vectors have Liouville measure zero. A compact simply connected modification region cannot contain a complete geodesic; this also treats the exotic connected-sum examples. Ergodicity then follows from the classical theorem on the regular set.