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Using tempered Følner sequences one may show a pointwise ergodic theorem for amenable groups.

(see http://www.aimsciences.org/journals/pdfsnews.jsp?paperID=2413&mode=full)

Is there a similar result for semigroups? Where by taking a special kind of Følner sequence you can achieve this. (Either for $L^{1}$ or $L^{2}$)

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  • $\begingroup$ I think something like this appears in Krengel's book. $\endgroup$
    – Anthony Quas
    Commented Dec 20, 2013 at 18:57

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Yes, see Theorem 4.5.3 in the Ph.D. thesis of Moy Easwaran: http://www.ams.org/mathscinet-getitem?mr=3103560

Here is Theorem 4.5.3 for ergodic actions:

Let $\Gamma$ be a discrete, countable, left-cancellative, left-amenable semigroup acting on $(X,\mathcal{B},\mu)$ via measure-preserving maps. Let $(F_N)_N$ be a tempered Følner sequence in $\Gamma$. If the action of $\Gamma$ is ergodic, then for all $f \in L^1(X,\mathcal{B},\mu)$, $$ \lim_{N \to \infty} \frac 1{|F_N|} \sum_{\gamma \in F_N} f(\gamma x) \longrightarrow \int f \ d\mu \qquad \mu\text{-a.e.}$$

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