Dynamics of vertical delay endomorphisms

A vertical delay endomorphism $F$ on $\mathbb{R}^k$, with $k\ge 2$, is the endomorphism associated to the difference equation $x_{n+k}=f(x_n,\cdots,x_{n+k-1})$, where the function $f$ is $C^2$ and its partial derivative of second order with respect to the first variable is bigger than every other partial derivative of second order. The main goal of this paper is to describe the dynamical behaviour of a huge class $\mathcal{F}$ of one-parameter families of vertical delay endomorphisms. We will prove that for any $\{F_\mu\}_{\mu\in\mathbb{R}}$ in $\mathcal{F}$ and every $|\mu|$ large enough, the nonwandering set $\Omega(F_\mu)$ of $F_\mu$, is either the empty set or an expanding Cantor set and the restriction of $F_{\mu}$ to $\Omega(F_\mu)$ is conjugated to the unilateral shift on two symbols.