The hierarchy of commuting maps related to a set-theoretical solution
of the quantum Yang-Baxter equation (Yang-Baxter map) is introduced. They can
be considered as dynamical analogues of the monodromy and/or transfer-matrices.
The general scheme of producing Yang-Baxter maps based on matrix factorisation is
discussed in the context of the integrability problem for the corresponding dynamical
systems. Some examples of birational Yang-Baxter maps coming from the theory of
the periodic dressing chain and matrix KdV equation are discussed.
This pre-print has been submitted, and accepted, to the journal, Physics Letters A. The definitive version: VESELOV, A.P.,2003. Yang-Baxter maps and integrable dynamics. Physics Letters A, 314(3),pp. 214-221, is available at: http://www.sciencedirect.com/science/journal/03759601.