On Mon, Dec 09, 2002 at 07:31:23PM -0600, James Propp wrote:
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". ...
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)
This hint led me to a different explanation of why this relation is true. There is a canonical source of period 5 phenomena: Consider the triangulations of a pentagon. There a 5 triangulations, and if you connect two triangulations that share a triangle, you get a 5-cycle. (Any two adjacent triangulations in this cycle are related by switching one diagonal of a quadrilateral for the other.) This same 5-cycle appears in many guises, in particular as the pentagon relation in category theory, related to the two diffrent ways to reparenthesize a(b(cd)) to ((ab)c)d: a(b(cd)) = a((bc)d) = (a(bc))d = ((ab)c)d a(b(cd)) = (ab)(cd) = ((ab)c)d To relate this to the problem at hand, consider 5 points in the complex plane. Up to Moebius transformations, the positions are determined by two cross ratios. A triangulation determines a choice of cross-ratios (one per diagonal): For each diagonal, apply a Moebius transformation to put the two endpoints at 0 and infinity; if the two adjacent vertices go to -x and y, the cross-ratio is defined to be y/x. It is easy to compute how the cross-ratios change when you flip a diagonal. They change by (l,m) -> (1/m, l(m+1)) which is related (by inverting the first component) to the expressions that Jim gave. Again, since switching the diagonals 5 times gets you back where you started, this transformation must be periodic with period 5. The reason the comments about tropical expressions ran a bell for me is that if you take this limit of the expressions for cross-ratios, you get corresponding expressions for transformations of curves on surfaces, which is something I've been thinking about a lot recently. One significant fact in this context is that the expressions you get (for any of the recurrences that Jim suggested) are a sum of monomials, or, equivalently, the PL functions are convex: they are the max of a number of alternatives. I don't yet have any geometric interpretation of the other recurrences, of periods six or eight. I'm still working on it... Best, Dylan