Invariant Curves and Focal Points in a Lyness Iterative Process

We investigate the properties of recurrence of the type xn+1 = (a+ ∑k−2 i=0 xn−i)/xn−(k−1), known as Lyness iterations from [Lyness, 1942, 1945, 1961] and recently analyzed by several authors in the case a > 0, see e.g. [Kocic et al., 1993; Csornyei & Laczkovich, 2000]. We reconsider Lyness recurrences at the light of some recent results on iterated maps with denominator, given in [Bischi et al., 1999a], where new kinds of singularities, such as focal points and prefocal curves, have been defined. In this paper, in particular, we give an answer to one of the open problems proposed in [Kocic & Ladas, 1993, pp. 141] concerning the dynamic behavior of Lyness recurrences for a < 0. We also give some new results in the case a > 0, and we improve a previous result on Lyness “periodic recurrences”.