Abstract
We study quadratic skew-products with parameters driven over piecewise expanding and Markov interval maps with countable many inverse branches, a generalization of the class of maps introduced by Viana. In particular, we construct a class of multi-dimensional non-uniformly expanding attractors that exhibit both critical points and discontinuities and prove existence and uniqueness of an SRB (Sinai–Ruelle–Bowen) measure with stretched-exponential decay of correlations, stretched-exponential large deviations and satisfying some limit laws. Moreover, generically such maps admit the coexistence of a dense subset of points with negative central Lyapunov exponent together with a full Lebesgue measure subset of points which have positive Lyapunov exponents in all directions. Finally, we discuss the existence of SRB measures for skew-products associated with hyperbolic parameters by the study of fibred hyperbolic maps.
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