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We present a tropical geometric description of a piecewise linear map whoseinvariant curve is a concave polygon. In contrast to convex polygons, a concaveone is not directly related to tropical geometry; nevertheless the descriptionis given in terms of the addition formula of a tropical elliptic curve. We showthat the map is arising from a pair of tropical elliptic pencils each member ofwhich is the invariant curve of the ultradiscrete QRT map.

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