  The paper is motivated by E. Akin's book about dynamical systems and closed
relations [A], and by J. Kennedy's and G. Erceg's recent paper about the
entropy of closed relations on closed intervals [EK]. In present paper, we
introduce the entropy of a closed relation G on any compact metric space X and
show its basic properties. We also introduce when such a relation G generates a
periodic point or finitely generates a Cantor set. Then we show that periodic
points, finitely generated Cantor sets, Mahavier products and the entropy of
closed relations are preserved by topological conjugations. Among other things,
this generalizes the well-known results about the topological conjugacy of
continuous mappings. Finally, we prove a theorem, giving sufficient conditions
for a closed relation G on [0,1] to have a non-zero entropy. Then we present
various examples of closed relations G on [0,1] such that (1) the entropy of G
is non-zero, (2) no periodic point or exactly one periodic point is generated
by G, and (3) no Cantor set is finitely generated by G.
