# On some ergodic properties for continuous and affine functions

• Volume: 28, Issue: 3, page 209-215
• ISSN: 0373-0956

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## Abstract

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Two problems posed by Choquet and Foias are solved:(i) Let $T$ be a positive linear operator on the space $C\left(X\right)$ of continuous real-valued functions on a compact Hausdorff space $X$. It is shown that if ${n}^{-1}{\sum }_{r=0}^{n-1}{T}^{r}1$ 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 $A\left(K\right)$ of continuous affine real-valued functions on $K$, such that$\mathrm{inf}\left\{\left({T}^{n}1\right)\left(x\right):n\ge \right\}<1$for each $x$ in ${\partial }_{\ell }K$, but $\parallel {T}^{n}1\parallel$ does not converge to 0.

## How to cite

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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 -

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