Title: Some integrable maps and their Hirota bilinear forms,
Abstract: We introduce a two-parameter family of birational maps, which reduces to a
family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when
one of the parameters is set to zero. The study of the singularity confinement
pattern for these maps leads to the introduction of a tau function satisfying a
homogeneous recurrence which has the Laurent property, and the tropical (or
ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic
degree growth found empirically by Demskoi et al. We prove that the tau
function also satisfies two different bilinear equations, each of which is a
reduction of the Hirota-Miwa equation (also known as the discrete KP equation,
or the octahedron recurrence). Furthermore, these bilinear equations are
related to reductions of particular two-dimensional integrable lattice
equations, of discrete KdV or discrete Toda type. These connections, as well as
the cluster algebra structure of the bilinear equations, allow a direct
construction of Poisson brackets, Lax pairs and first integrals for the
birational maps. As a consequence of the latter results, we show how each
member of the family can be lifted to a system that is integrable in the
Liouville sense, clarifying observations made previously in the original DTKQ
case.