  Let (A,H,D) be a spectral triple, namely: A is a C*-algebra, H is a Hilbert
space on which A acts and D is a selfadjoint operator with compact resolvent
such that the set of elements of A having a bounded commutator with D is dense.
A spectral metric space, the noncommutative analog of a complete metric space,
is a spectral triple (A,H,D) with additional properties which guaranty that the
Connes metric induces the weak*-topology on the state space of A. A
*-automorphism respecting the metric defined a dynamical system. This article
gives various answers to the question: is there a canonical spectral triple
based upon the crossed product algebra AxZ, characterizing the metric
properties of the dynamical system ? If $\alpha$ is the noncommutative analog
of an isometry the answer is yes. Otherwise, the metric bundle construction of
Connes and Moscovici is used to replace (A,$\alpha$) by an equivalent dynamical
system acting isometrically. The difficulties relating to the non compactness
of this new system are discussed. Applications, in number theory, in coding
theory are given at the end.
