Degree growth of skew pentagram maps

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.

💡 Research Summary

This paper conducts a rigorous algebraic analysis of the integrability of skew pentagram maps, a generalization of the classical integrable pentagram map introduced by Khesin and Soloviev. The central tool is the introduction of a notion of dynamical degree for lattice maps—shift-equivariant self-maps of infinite sequences of projective points defined by local intersection rules, such as (P^N)^Z.

The main results are dichotomous:

1. Equal-length maps are integrable from the degree-growth perspective. When the intersecting diagonals have the same combinatorial length (|b-a| = |d-c|), the corresponding skew pentagram map T_{a,b,c,d} has a dynamical degree λ₁ = 1. This signifies polynomial growth in the algebraic complexity of iterates. The proof embeds these maps into the Schwarzian octahedron recurrence, a known integrable system with polynomial degree growth, leveraging techniques from earlier work on the classical pentagram map.
2. Infinitely many truly skew maps exhibit maximal chaos. For maps where the diagonal lengths differ, the authors prove that there are infinitely many parameter choices (a, b, c, d) for which the dynamical degree is 4. Since each new vertex depends linearly on four previous vertices, 4 is the maximum possible algebraic degree for the local rule, providing an upper bound. To prove this bound is achieved (a lower bound), the paper focuses on a specific example, T_{0,2,1,4}. The key innovation is an “entropy sandwich” strategy: instead of computing the dynamical degree on the high-dimensional space of all polygons, the map is restricted to the invariant subspace of closed octagons with 4-fold rotational symmetry. This moduli space is a surface. By constructing an algebraically stable model for the induced map T_RS on this surface via blow-ups, the authors compute its dynamical degree to be exactly 4. This implies the original lattice map T_{0,2,1,4} also has dynamical degree 4. Using modular arithmetic, this result extends to an infinite family of parameters congruent to (0,2,1,4) modulo 8.

Further consequences are derived:

• Arithmetic Entropy: Applying recent results on the Kawaguchi-Silverman conjecture, the authors show that the arithmetic degree (exponential growth rate of heights of rational points) equals the dynamical degree. This theoretically confirms Khesin-Soloviev’s experimental observation of exponential height growth for truly skew maps versus polynomial growth for equal-length maps.
• Topological Entropy & Fibrations: A dynamical degree greater than 1 implies positive topological entropy. Moreover, it rules out the existence of a preserved rational fibration, a property associated with integrability. Thus, the truly skew map T_{0,2,1,4} (and its infinite family) is shown to lack such a fibration.
• A Conjecture: Based on the evidence, the authors conjecture that all truly skew pentagram maps have dynamical degree 4.

In summary, the paper establishes a sharp algebraic boundary between integrable (equal-length) and non-integrable (truly skew) pentagram maps by analyzing their degree growth. It introduces a versatile framework for dynamical degrees of lattice maps and provides concrete calculations demonstrating maximal algebraic entropy in a broad class of examples, thereby advancing the classification of integrable systems within this geometric family.