Robert Bryant: the Geometry of Periodic Equi-Areal Sequences

Robert Bryant, Duke University
29 January 2015

Dr. Bryant began with the observation that the map $F(x,y)=(y,\frac{y+1}{x})$ is five-periodic; that is, $F\circ F\circ F\circ F\circ F$ is the identity, at least on its domain of definition. This domain is the complement of five lines, and the map $F$ can be easily described by how it permutes the twelve regions those five lines cut the plane into.

This is really the statement that the recurrence relation $y_{k+1}=\frac{y_k+1}{y_{k-1}}$ is five-periodic for any choice of initial data. Also, observe that if we set $f_k=\log y_k$ , we get the recurrence $f_{k+1}+f_{k-1}=\log(1+\exp(f_k))$ , which implies in turn that $df_k\wedge df_{k+1}=df_{k+1}\wedge df_{k+2}$ . That is, any adjacent pair of the $f_k$ give the same area form on the domain of the $f_k$ . We call such a sequence of functions an equiareal sequence.

It turns out that periodic equiareal sequences are hard to come by. The example coming from $F$ that we just discussed is, in fact, the only such five-periodic sequence. There aren’t any four-periodic sequences, and again only one three-periodic sequence.

Examples of five-, six-, and eight-periodic sequences can be constructed from the cluster algebras $A_2,B_2,G_2$ respectively. Are there others? What algebra do they reflect? What can this tell us about the geometry of the surfaces whose area forms are given by the $df_k\wedge df_{k+1}$ ?

Note: Dr. Bryant is an NCSU alum!