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

We consider a one-parameter family of maps Fλ on Rm×Rn with the singular map F0 having one of the two forms (i) F0(x, y) = (f(x), g(x)), where f:Rm→Rm and g:Rm→Rn are continuous, and (ii) F0(x, y) = (f(x), g(x, y)), where f:Rm→Rm and g:Rm×Rn→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 Fλ has a positive topological entropy for all λ close enough to 0