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We consider positive linear operators acting on C(K), where K is a metrizable Bauer simplex. For such an operator L we investigate the limit of the iterates L m , when m → ∞. Qualitative results and rates of convergence are obtained. The general results are illustrated by examples involving classical operators. Mathematics Subject Classification (2010): 41A36, 46A55.doi:10.24193/subbmath.2019.3.04 fatcat:53qp4qyyajeifhla727dbz34c4