Ergodic theorem for flows on closed surfaces

The flows on closed surfaces can either be topologically transitive or metrically transitive. The Morse conjecture for flows on closed oriented surfaces as stated in Theorem A is discussed. Theorem A assumes that if a C1-flow ft on M has only finitely many equilibrium states, then, the flow ft is metrically transitive if it is topologically transitive. Then, as an immediate consequence, an improvement of the classical ergodic theorem of Birkhoff type (Theorem B) is discussed. In Theorem B, ??t is assumed to be an area-preserving flow on M having only finitely many equilibrium states. If ??t is a topologically transitive flow, then, the Birkhoff ergodic theorem is valid.EI6669-6763