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Abstract

On a compact metric space $X$ one defines a transition system to be a lower semicontinuous map $X\to 2^X$. It is known that every Markov operator on $C(X)$ induces a transition system on $X$ and that commuting of Markov operators implies commuting of the induced transition systems. We show that even in finite spaces a pair of commuting transition systems may not be induced by commuting Markov operators. The existence of trajectories for a pair of transition systems or Markov operators is also investigated.