# On some ergodic properties for continuous and affine functions

Annales de l'institut Fourier (1978)

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

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topBatty, 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 < 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 < 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 -

## References

top- [1] G. CHOQUET and C. FOIAS, Solution d'un problème sur les itérés d'un opérateur positif sur C (K) et propriétés de moyennes associées, Ann. Inst. Fourier (Grenoble), 24, no. 3 & 4 (1975), 109-129. Zbl0303.47004MR53 #11017
- [2] J. DIXMIER, Les C*-algèbres et leurs représentations, 2nd ed., Gauthier-Villars, Paris, 1969. Zbl0174.18601

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