  In the last decades the problem of metastability has been attacked on
rigorous grounds via many different approaches and techniques which are briefly
reviewed in this paper. It is then useful to understand connections between
different point of views. In view of this we consider irreducible, aperiodic
and reversible Markov chains with exponentially small transition probabilities
in the framework of Metropolis dynamics. We compare two different cycle
decompositions and prove their equivalence.
