Hamiltonian Systems

The pentagram map acts on projective polygons by connecting their nearest diagonals and successively intersecting them. Convex polygons are preserved under this map and have precompact orbits in the projective moduli space. The deeper diagonal map $T_k$ generalizes the pentagram map by intersecting consecutive diagonals that are formed by vertices $k$-clicks apart. For $k \geq 3$, $T_k$ doesn't preserve convexity (or even embeddedness) of closed polygons.

However, we found that a certain subset of the so-called twisted polygons is preserved under $T_k$. We call them $k$-spirals, where the name is motivated by the choice of representatives that resembles inward spirals on the affine patch. There are two types of $k$-spirals depending on the configuration of the four vertices $i$, $i+1$, $i+k$, $i+k+1$. We show that $T_k$ preserves each of the two types of $k$-spirals. In particular, we show that the corner invariants parameterization of twisted polygons partitions the $3$-spirals nicely into four compartments, where the boundary of the partition resembles the shape of a tic-tac-toe board. This allows us to show that the $3$-spirals have precompact $T_3$ orbits, indicating that the ($T_k$, $k$-spiral) pair is a nice geometric analog to the ($T_2$, convex) pair.

The talk will be via Zoom - please see link and password above.

https://arxiv.org/abs/2412.15561