  We compare two different models of noncommutative geometry of the cyclotomic
tower, both based on an arithmetic algebra of functions of roots of unity and
an action by endomorphisms, the first based on the Bost-Connes quantum
statistical mechanical system and the second on the Habiro ring, where the
Habiro functions have, in addition to evaluations at roots of unity, also full
Taylor expansions. Both have compatible endomorphisms actions of the
multiplicative semigroup of positive integers. As a higher dimensional
generalization, we consider a crossed product ring obtained using Manin's
multivariable generalizations of the Habiro functions and an action by
endomorphisms of the semigroup of integer matrices with positive determinant.
We then construct a corresponding class of multivariable Bost-Connes
endomotives, which are obtained geometrically from self maps of higher
dimensional algebraic tori, and we discuss some of their quantum statistical
mechanical properties. These multivariable Bost-Connes endomotives are
universal for (torsion free) Lambda-rings, compatibly with the Frobenius
action. Finally, we discuss briefly how Habiro's universal
Witten--Reshetikhin--Turaev invariant of integral homology 3-spheres may relate
invariants of 3-manifolds to gadgets over the field with one element and
semigroup actions on homology 3-spheres to endomotives.
