I asked earlier about the Lyness sequence, given by the recurrence L(n+1) = (L(n)+1)/L(n-1). Regardless of the initial conditions, a sequence satisfying this recurrence will be periodic with period 5; this is easy to verify, but I wanted to know "why". Some illumination came from reading parts of the paper "Y-systems and generalized associahedra" by Sergey Fomin and Andrei Zelevinsky, available over the web at http://www.math.lsa.umich.edu/~fomin/Papers/ (to appear in Annals of Mathematics), which Michael Kleber and I looked at. First, we may switch over to thinking about composition of the two rational functions F and G where F(s,t) = (s,(s+1)/t) and G(s,t) = ((t+1)/s,t) . F and G are both involutions, but if we take all alternating compositions ...(F(G(F(...(s,t)...)))... we get a group of 10 operations. In particular, G-compose-F is of order 5. Second, noting that the expressions for F and G are subtraction-free, we may replace the maps F and G their "tropical analogues" f and g, replacing the constant 1 by the constant 0 and replacing the arithmetical operations +, *, and / by max, +, and -: f(s,t) = (s,max(s,0)-t) g(s,t) = (max(t,0)-s,t) Note that f and g are continuous piecewise linear operations on the plane. Each of f,g is of order 2, but their composition g-compose-f has order 5. Moreover, the orbit of (1,0) under the action of the composed map is (1,0), (0,1), (0,-1), (1,1), and (-1,0) which we can recognize as being related to the Laurent polynomials x, y, (y+1)/x, (x+y+1)/xy, (x+1)/y from the Lyness sequence when they are written as (1) / x^{-1} y^{0} (1) / x^{0} y^{-1} (1+y) / x^{1} y^{0} (1+x+y)/ x^{1} y^{1} (1+x) / x^{0} y^{1} To see what's going on with f and g geometrically, it's most helpful to think of (1,0) and (0,1) as making an angle of 120 degrees, and to divide the plane up into 60-degree sectors. However, the maps f and g do not permute these sectors (if they did, then their composition could not have order 5 without forcing f and/or g to be discontinuous). Instead, some sectors are mapped into sectors, some sectors are expanded into a union of two adjacent sectors, and some sectors are contracted into a part of a sector. A similar situation prevails for the 6-fold periodic recurrence and for the 8-fold periodic recurrence, and there is a Lie-theoretic story for what's going, involving the root-lattice. E.g., for the original 5-fold case, if you examine the list above you can check that if we throw out the first two denominators (the ones with negative exponents), the resulting vectors (1,0), (0,1), and (1,1) are the positive roots of the Lie algebra A_2 (relative to the basis of simple roots). This happens much more generally. By the way: Does anyone see a way to combine the 10-fold Lyness action with the 6-fold rotation action on the 6 sectors to get the alternating group A_5, or anything like it? Low-dimensional piecewise-linear maps might give faithful representations of interesting groups whose linear representations are all higher-dimensional. Jim Propp