Degree growth of skew pentagram maps

TL;DR

This paper investigates the degree growth of skew pentagram maps, revealing some exhibit exponential growth and lack preserved fibrations, contrasting with integrable pentagram maps.

Contribution

It introduces a notion of dynamical degree for lattice maps and demonstrates that certain skew pentagram maps have exponential degree growth and dynamical degree 4.

Findings

Equal-length pentagram maps have dynamical degree 1.

Skew pentagram maps can have dynamical degree 4.

Some skew maps exhibit exponential degree growth.

Abstract

Skew pentagram maps act on polygons by intersecting diagonals of different lengths. They were introduced by Khesin-Soloviev in 2015 as conjecturally non-integrable generalizations of the pentagram map, a well-known integrable system. In this paper, we show that certain skew pentagram maps have exponential degree growth and no preserved fibration. To formalize this, we introduce a general notion of first dynamical degree for lattice maps, or shift-invariant self-maps of $(\mathbb{P}^N)^\mathbb{Z}$. We show that the dynamical degree of any equal-length pentagram map is 1, but that there are infinitely many skew pentagram maps with dynamical degree 4.