A characterization of kk-cycles

We study global periodicity for the difference equation of order ll given by xn+l=f(xn+l−1,xn+l−2,…,xn)xn+l=f(xn+l−1,xn+l−2,…,xn), where f:(0,+∞)l→(0,+∞)f:(0,+∞)l→(0,+∞) is a continuous map, l∈Z+l∈Z+. Our main results are the following. We prove that if any solution of the equation is periodic, then there is a minimal k∈Nk∈N such that the period of any solution divides kk (and therefore ff is called a kk-cycle). In addition, if l=2l=2, then for any k>2k>2 there are, up to conjugacy, only a kk-cycle. Finally, if l>2l>2 and ff gives a (l+1)(l+1)-cycle, then ff is topologically conjugate to:
• xn+l=1xn⋅xn+1⋅⋯⋅xn+l−1, if ll is even.
• The previous equation or xn+l=∏j=1(l+1)/2xn+2j−2∏j=1(l−1)/2xn+2j−1, if ll is odd. Our results solve some open questions from [J. S. Cánovas, A. Linero and G. Soler, On global periodicity of difference equations, Taiwanese J. Math. (in press)] and [M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations. With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, FL, 2002].