Building Expansion for Generalizations of Viana Maps

We study a family of skew-products of smooth functions having a unique critical point of degree \(D\ge 2\) $D\ge 2$ over a strongly expanding map of the circle and prove that these systems admit two positive Lyapunov exponents. This extends an analogous result of Viana who considered, in the seminal paper (Viana in Inst Hautes Études Sci Publ Math 85:63–96, 1997), the quadratic case \(D=2\) $D=2$ .