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A generalization of Zeeman’s family

Michał Sierakowski — 1999

Fundamenta Mathematicae

E. C. Zeeman [2] described the behaviour of the iterates of the difference equation x n + 1 = R ( 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 k = 1 , Q = x n - 1 and R = x n + α , 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.

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