Endomotives of toric varieties

We construct endomotives associated to toric varieties, in terms of the decomposition of a toric variety into torus orbits and the action of a semigroup of toric morphisms. We show that the endomotives can be endowed with time evolutions and we discuss the resulting quantum statistical mechanical systems. We show that, in particular, one can construct a time evolution related to the logarithmic height function. We discuss relations to F1-geometry.