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

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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.