Rectangle exchange maps (REMs) are higher dimensional generalizations
of interval exchange maps which have been well-studied for more than fourty
years. We study REMs that arise from cut-and-project schemes on lattices
associated to cubic Pisot numbers. We prove that these REMs are minimal
and self-induced. Moreover, these REMs are parametrized by matrices in
SL(7,Z) whose leading non-zero eigenvalues are Pisot numbers. Via matrix
products, we identify a family of renormalizable REMs whose parameter space
is a four-dimensional Cantor set. We will give a symbolic encoding of the
dynamical system on the parameter space. This is joint work with Ian Alevy
and Richard Kenyon.