Standardness of automorphisms of transposition of intervals and fluxes on surfaces

In this article we shall prove a new necessary and sufficient condition for automorphisms to be standard, from which we shall deduce the standardness of an automorphism of transposition of intervals with respect to any continuous Borel invariant ergodic measure, and the standardness of the flux of the class $C^1$ on a two-dimensional compact variety with a finite number of stationary points and separatrices, with respect to any Borel invariant ergodic measure whose carrier contains an open set.