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Invariant means and iterates of mean-type mappings
A classical result states that for two continuous, strict means M, N : I 2 → I (I is an interval) there exists a unique (M, N )-invariant mean K : I 2 → I, i.e. such a mean that K • (M, N ) = K and, moreover, the sequence of iterates ((M, N ) n ) ∞ n=1 converge to (K, K) pointwise. Recently it was proved that continuity assumption cannot be omitted in general. We show that if K is a unique (M, N )-invariant mean then, without continuity assumption, (M, N ) n → (K, K). Mathematics Subject Classification. 26E60, 26D15.doi:10.1007/s00010-019-00668-3 fatcat:mj4y6qj4uvby5edgnmnt55qosi