  We present a solvable two-dimensional piecewise linear chaotic map which
arises from the duplication map of a certain tropical cubic curve. Its general
solution is constructed by means of the ultradiscrete theta function. We show
that the map is derived by the ultradiscretization of the duplication map
associated with the Hesse cubic curve. We also show that it is possible to
obtain the nontrivial ultradiscrete limit of the solution in spite of a problem
known as "the minus-sign problem."
