On some ergodic properties for continuous and affine functions

Batty, Charles J. K.. "On some ergodic properties for continuous and affine functions." Annales de l'institut Fourier 28.3 (1978): 209-215. <http://eudml.org/doc/74372>.

@article{Batty1978,
abstract = {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 $n^\{-1\}\sum ^\{n-1\}_\{r=0\} T^r1$ 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\begin\{\}\{\rm inf\}\lbrace (T^n1)(x) : n\ge \rbrace &lt; 1\end\{\}for each $x$ in $\partial _\ell K$, but $\Vert T^n1\Vert $ does not converge to 0.},
author = {Batty, Charles J. K.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {3},
pages = {209-215},
publisher = {Association des Annales de l'Institut Fourier},
title = {On some ergodic properties for continuous and affine functions},
url = {http://eudml.org/doc/74372},
volume = {28},
year = {1978},
}

TY - JOUR
AU - Batty, Charles J. K.
TI - On some ergodic properties for continuous and affine functions
JO - Annales de l'institut Fourier
PY - 1978
PB - Association des Annales de l'Institut Fourier
VL - 28
IS - 3
SP - 209
EP - 215
AB - 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 $n^{-1}\sum ^{n-1}_{r=0} T^r1$ 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\begin{}{\rm inf}\lbrace (T^n1)(x) : n\ge \rbrace &lt; 1\end{}for each $x$ in $\partial _\ell K$, but $\Vert T^n1\Vert $ does not converge to 0.
LA - eng
UR - http://eudml.org/doc/74372
ER -