Entropy along expanding foliations

The (measure-theoretical) entropy of a diffeomorphism along an expanding invariant foliation is the rate complexity generated by leaves foliation. We prove that this number varies upper semi-continuously with (C1 topology), measure (weak* topology) and itself in suitable sense. This has several important consequences. For one thing, it implies set Gibbs u-states C1+ partially hyperbolic diffeomorphisms semi-continuous function map C1 topology. Another consequence sets mostly contracting or center are open. New examples provided, existence physical measures for residual subset discussed. also provide new class robustly transitive diffeomorphisms: every C2 volume preserving, accessible dimensional non-vanishing exponent (among neighborhood which not necessarily preserving).