A generalization of Zeeman’s family

E. C. Zeeman [2] described the behaviour of the iterates of the difference equation ${\displaystyle x_{n+1}=R\frac{x_n,x_{n-1},...,x_{n-k}}{Q}(x_n,x_{n-1},...,x_{n-k})}$, n ≥ k, R,Q polynomials in the case ${\displaystyle k=1,Q=x_{n-1}}$ and ${\displaystyle R=x_n+\alpha }$, ${\displaystyle x_1,x_2}$ positive, α nonnegative. We generalize his results as well as those of Beukers and Cushman on the existence of an invariant measure in the case when R,Q are affine and k = 1. We prove that the totally invariant set remains residual when the coefficients vary.