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On some ergodic properties for continuous and affine functions

Charles J. K. Batty (1978)

Annales de l'institut Fourier

Two problems posed by Choquet and Foias are solved:(i) Let T be a positive linear operator on the space C ( X ) of continuous real-valued functions on a compact Hausdorff space X . It is shown that if n - 1 r = 0 n - 1 T r 1 converges pointwise to a continuous limit, then the convergence is uniform on X .(ii) An example is given of a Choquet simplex K and a positive linear operator T on the space A ( K ) of continuous affine real-valued functions on K , such that inf { ( T n 1 ) ( x ) : n } < 1 for each x in K , but T n 1 does not converge to 0.

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