Hyperbolic invariant sets of the real generalized Hénon maps

In this paper, the conditions under which there exists a uniformly hyperbolic invariant set for the generalized Hénon map F(x, y) =  (y, ag(y) − δx) are investigated, where g(y) is a monic real-coefficient polynomial of degree d ⩾ 2, a and δ are non-zero parameters. It is proved that for certain parameter regions the map has a Smale horseshoe and a uniformly hyperbolic invariant set on which it is topologically conjugate to the two-sided fullshift on two symbols, where g(y) has at least two different non-negative or non-positive real zeros, and ∣a∣ is sufficiently large. Moreover, it is shown that if g(y) has only simple real zeros, then for sufficiently large ∣a∣, there exists a uniformly hyperbolic invariant set on which F is topologically conjugate to the two-sided fullshift on d symbols.