Thick hyperbolic repelling invariant Cantor sets and wild attractors

Abstract

Abstract Let D be the set of β ∈ ( 1 , 2 ] such that f β is a symmetric tent map with finite critical orbit. For β ∈ D , by operating Denjoy-like surgery on f β , we constructed a C 1 unimodal map g ˜ β admitting a thick hyperbolic repelling invariant Cantor set which contains a wild attractor. The smoothness of g ˜ β is ensured by the effective estimation of the preimages of the critical point as well as the prescribed lengths of the inserted intervals. D is dense in ( 1 , 2 ] , and g ˜ β can not be C 1 + α because the hyperbolic repelling invariant Cantor set of C 1 + α map has Lebesgue measure equal to zero.