Artur Avila and Marcelo Viana have obtained a non-linear version of a famous result on the positivity of Lyapunov exponents of linear cocycles (initially due to Furstenberg), which has lead to many consequences for volume-preserving partially hyperbolic diffeomorphisms whose center leaves are compact.

In a joint work with Mauricio Poletti we extend it to a setting where the center leaves are non compact and we derive new consequences.