Calculating algebraic entropies: an express method

Abstract

We describe a method for investigating the integrable character of a given three-point mapping, provided that the mapping has confined singularities. Our method, dubbed "express", is inspired by a novel approach recently proposed by R.G. Halburd. While the latter aims at computing the exact degree growth of a given mapping based on the structure of its singularities, we content ourselves with obtaining an answer as to whether a given system is integrable or not. We present several examples illustrating our method as well as its limitations. We also compare the present method to the full-deautonomisation approach we recently introduced.

Singularity confinement as an integrability criterion

“…This is not surprising as the original relation (13) in the express method obviously corresponds to the action of the pullback ϕ˚, which is related to the push-forward by ϕ˚" pϕ´1q˚, on the curves C j . Moreover, the construction of the equations (55"56) itself also corresponds perfectly to the direct construction of the characteristic polynomials (14), (15) and (16) from the singularity patterns, as given in (17).…”

“…This is a possibility we already pointed out when we introduced the express method. In [17] we give an example of a discrete Painlevé equation with E p1q 8 symmetry, that has 8 short open singularity patterns. In fact, these patterns are too short to yield any useful relations that would allows us to implement the express method (or, for that matter, the present method).…”

“…In this section we shall explain the method we proposed in [17] for obtaining the dynamical degree of a confining mapping, directly from its singularity patterns, on two examples: mapping (1) from the introduction and a simple multiplicative QRT mapping which, when deautonomized, leads to a qPI equation. The full, Halburd-style, analysis of these mappings which yields the exact degrees of the iterates of the mapping, is given in Appendix 1.…”

Calculating the algebraic entropy of mappings with unconfined singularities

“…An ingeniously simple method was introduced by Halburd in [12]. It was simplified by the present authors into what we call the express method [5].…”

“…Many systems also possess singularities in which a pattern keeps repeating for all iterations. We have dubbed these singularities 'cyclic' [5]. The existence of cyclic singularities is perfectly compatible with integrability and, as such, cyclic singularities are definitely not be to thought of as unconfined.…”