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?
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asked Oct 27, 2012 at 17:09
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1 Answer
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Yes, this map is conjugate to an automorphism of $\mathbf{P}^2$. See
A. Beauville, J. Blanc, On Cremona transformations of prime order, 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.
answered Oct 27, 2012 at 17:51
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$\begingroup$ The reference is definitely helpful, especially the explicit formulas on page 3. However, I don't know how to explicitly see the map $(x:y:z) \mapsto (x(z-y):z(x-y):xz)$ (the birational transformation that Beauville and Blanc treat) as being conjugate to the map $(x:y:z) \mapsto (xy:(y+z)z:xz)$ (the projective version of the Lyness 5-cycle map). $\endgroup$ Commented Oct 29, 2012 at 16:33
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$\begingroup$ @James : Let $f : (x,y) \mapsto (y,\frac{y+1}{x})$ and $g : (x,y) \mapsto (1-y,1-\frac{y}{x})$. Then $g=hfh^{-1}$ with $h(x,y)=(-x,y+1)$. I didn't check this, but I knew the result had to be true since both maps are related to the functional equation of the dilogarithm :) By the way, the first versions of Beauville's article on arxiv use this functional equation to prove the result. $\endgroup$ Commented Oct 30, 2012 at 11:41
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$\begingroup$ @Francois: That answers my question. Thanks! $\endgroup$ Commented Nov 2, 2012 at 1:27