Invariance principle and rigidity of high entropy measures
  A deep analysis of the Lyapunov exponents, for stationary sequence of matrices going back to Furstenberg, for more general linear cocycles by Ledrappier and generalized to the context of non-linear cocycles by Avila and Viana, gives an invariance principle for invariant measures with vanishing central exponents. In this paper, we give a new criterium formulated in terms of entropy for the invariance principle and in particular, obtain a simpler proof for some of the known invariance principle results.   As a byproduct, we study ergodic measures of partially hyperbolic diffeomorphisms whose center foliation is 1-dimensional and forms a circle bundle. We show that for any such C2 diffeomorphism which is accessible, weak hyperbolicity of ergodic measures of high entropy implies that the system itself is of rotation type. 
Ali Tahzibi and Jiagang Yang
math.DS