The Lyness 5-cycle is the map that sends $(x,y)$ to $(y,z)$ with $z=(y+1)/x$. Leaving aside the set on which the map is not well-defined, the map is of order 5 (hence its name). Is there an algebraic map that conjugates the map to a rotation by 72 degrees?



Conjugating the Lyness 5-cycle into a rotation of the plane
Yes, this map is conjugate to an automorphism of $\mathbf{P}^2$. See


A. Beauville, J. Blanc, [*On Cremona transformations of prime order*](http://arxiv.org/abs/math/0402037), C.R. Acad. Sci. Paris 339 (2004), no4, 257-259.


See also T. de Fernex, *On planar Cremona maps of prime order*, Nagoya Math. Journal, Vol. 174 (2004), 1–28, which contains a classification of planar Cremona maps of prime order up to conjugation. According to Remark 1.3.4 therein, the fact above was already known to Iskovskikh.


