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 α 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,α) 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