POSITIVE TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF TOPOLOGICALLY CROSSING HOMOCLINICITY

Abstract

In this paper, we consider a one-parameter family F(lambda) of continuous maps on R(m) or R(m) x R(k) with the singular map F(0) having one of the forms (i) F(0)(x) = f(x); (ii) F(0)(x,y) = (f(x), g(x)), where g : R(m) -> R(k) is continuous, and (iii) F(0)(x; y) = (f(x), g(x,y)), where g : R(m)xR(k) -> R(k) is continuous and locally trapping along the second variable y. We show that if f : R(m) -> R(m) is a C(1) diffeomorphism having a topologically crossing homoclinic point, then F(lambda) has positive topological entropy for all lambda close enough to 0.