Topological entropy for multidimensional perturbations of snap-back repellers and one-dimensional maps

Abstract

We consider a one-parameter family of maps F(lambda) on R(m) x R(n) with the singular map F(0) having one of the two forms (i) F(0) (x, y) = (f (x), g(x)), where f : R(m) -> R(m) and g : R(m) -> R(n) are continuous, and (ii) F(0)( x, y) = (f (x), g(x, y)), where f : R(m) -> R(m) and g : R(m) x R(n) -> Rn are continuous and g is locally trapping along the second variable y. We show that if f is one-dimensional and has a positive topological entropy, or if f is high-dimensional and has a snap-back repeller, then F. has a positive topological entropy for all lambda close enough to 0.