  The theory of limiting modular symbols provides a noncommutative geometric
model of the boundary of modular curves that includes irrational points in
addition to cusps. A noncommutative space associated to this boundary is
constructed, as part of a family of noncommutative spaces associated to
different continued fractions algorithms, endowed with the structure of a
quantum statistical mechanical system. Two special cases of this family of
quantum systems can be interpreted as a boundary of the system associated to
the Shimura variety of $GL_2$ and an analog for $SL_2$. The structure of KMS
states for this family of systems is discussed. In the geometric cases, the
ground states evaluated on boundary arithmetic elements are given by pairings
of cusp forms and limiting modular symbols.
