Approximation of entropy on hyperbolic sets for one-dimensional maps and their multidimensional perturbations

We consider piecewise monotone (not necessarily, strictly) piecewise $C^2$ maps on the interval with positive topological entropy. For such a map $f$ we prove that its topological entropy $h_{top}(f)$ can be approximated (with any required accuracy) by restriction on a compact strictly $f$-invariant hyperbolic set disjoint from some neighborhood of prescribed set consisting of periodic attractors, nonhyperbolic intervals and endpoints of monotonicity intervals. By using this result we are able to generalize main theorem from [1] on chaotic behavior of multidimensional perturbations of solutions for difference equations which depend on two variables at nonperturbed value of parameter.