  We propose a construction for spectral triple on algebras associated with
subshifts. One-dimensional subshifts provide concrete examples Z-actions on
Cantor sets. The C*-algebra of this dynamical system is generated by functions
in C(X) and a unitary element u implementing the action. Building on ideas of
Christensen and Ivan, we give a construction of a family of spectral triples on
the commutative algebra C(X). There is a canonical choice of eigenvalues for
the Dirac operator D which ensures that [D,u] is bounded, so that it extends to
a spectral triple on the crossed product.
  We study the summability of this spectral triple, and provide examples for
which the Connes' distance associated with it on the commutative algebra is
unbounded, and some for which it is bounded. We conjecture that our results on
the Connes distance extend to the spectral triple defined on the noncommutative
algebra.
