A classical result states that for two continuous, strict means \(M,\,N :I^2 \rightarrow I\) (I is an interval) there exists a unique (M, N)-invariant mean \(K :I^2 \rightarrow I\), i.e. such a mean that \(K \circ (M,N)=K\) and, moreover, the sequence of iterates \(((M,N)^n)_{n=1}^\infty \) 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, \((M,N)^n \rightarrow (K,K)\).

中文翻译：

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均值映射的不变均值和迭代

一个经典的结果表明，对于两个连续的严格均值\（M，\，N：I ^ 2 \ rightarrow I \）（I是一个区间），存在唯一的（M，  N）不变均值\（K：I ^ 2 \ rightarrow I \），即\（K \ circ（M，N）= K \）的平均值，此外，迭代序列（（（（M，N）^ n）_ {n = 1} ^ \ infty \）逐点收敛到（K，  K）。最近，证明了连续性假设通常不能被忽略。我们证明如果K是唯一的（M，  N）不变均值，则在不连续性假设的情况下\（（M，N）^ n \ rightarrow（K，K）\）。