An ultradiscrete integrable map arising from a pair of tropical elliptic pencils

We present a tropical geometric description of a piecewise linear map whose invariant curve is a concave polygon. In contrast to convex polygons, a concave one is not directly related to tropical geometry; nevertheless the description is given in terms of the addition formula of a tropical elliptic curve. We show that the map arises from a pair of tropical elliptic pencils, each member of which is the invariant curve of an ultradiscrete QRT map. ► We present a tropical geometric description of a piecewise linear map. ► The map is integrable and its invariant curve is a concave nonagon. ► The map is reformulated in terms of the addition formula of a tropical elliptic curve. ► The general solution to the map is given by using the ultradiscrete theta function.