Positive Exponents for Random Products of Conservative Surface Diffeomorphisms and Some Skew Products

In this paper we show that a “typical” random product of conservative surface diffeomorphism has positive Lyapunov exponents. We prove that for any compact oriented surface S, any $$r\ge 1$$ , and any $$d\ge 2$$ , there exists a $$C^1$$ -open and $$C^1$$ -dense subset of $${\text {Diff}}^r_{{\text {vol}}}(S)^d$$ such that if $$(f_1, \ldots , f_d)$$ belongs to this subset, the random product generated by them has positive Lyapunov exponents. Our proof also allows us to deal with more general skew products, for example skew products with a volume preserving Anosov diffeomorphism on the basis, or with a subshift of finite type on the basis preserving a measure with product structure. In these cases we prove the $$C^1$$ -density and $$C^r$$ -openness of the existence of positive Lyapunov exponents.