Coexistence of invariant sets with and without SRB measures in Henon family

Abstract

Let {f(a,b)} be the (original) Henon family. In this paper, we show that, for any b near 0, there exists a closed interval J(b) which contains a dense subset J' such that, for any a is an element of J', f(a,b) has a quadratic homoclinic tangency associated with a saddle fixed point of f(a,b) which unfolds generically with respect to the one-parameter family {f(a,b)}(a is an element of Jb). By applying this result, we prove that J(b) contains a residual subset A(b)((2)) such that, for any a is an element of A(n)((2)), f(a,b) admits the Newhouse phenomenon. Moreover, the interval Jb contains a dense subset (A) over tilde (b) such that, for any a is an element of (A) over tilde (b), f(a,b) has a large homoclinic set without SRB measure and a small strange attractor with SRB measure simultaneously.