Yang-Baxter maps and integrable dynamics

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.