On the abundance of chaotic behavior for generic one-parameter families of maps

Abstract

In this paper, we want to show the abundance of chaotic systems with absolutely continuous probability measures in the generic regular family with perturbable points. More precisely, we prove that iffa:I → I, a ∈P is a regular family satisfying some conditions described in the next section, then there exists a Borel set Ω ⊂P of positive Lebesgue measure such that for everya ∈ Ω,fa admits an absolutely continuous invariant probability measure w.r.t. the Lebesgue measure. The idea of proof in this paper, as compared with that shown in [1] and [7], follows a similar line.