The classical Kac's lemma says the following.

Let $(X,\mu)$ be a probability space and $T$ a measure preserving transformation. Assume $A\subset X$ has positive measure. Then $$\sum_{k\ge 1} k\mu(A_k)=1,$$ where $A_k$ denotes the set of points in $A$ that return to $A$ by first time after exactly $k$ iterates of $T$.

Question: Is there an 'analog' of Kac's lemma for amenable group actions? with suitably choice of folner sequence and dealing with the (a priori) lack of disjointness that the sets $A_k$'s propagate. Of course I am not expecting the (suitable) weigthed sum to have value 1, but something finite.

It sounds like a natural question and I am not sure if it's a known fact.

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    $\begingroup$ Before considering a general amenable group, can you make sense of your question for $\mathbb{Z}^2$? $\mathbb{Z}/2\ltimes\mathbb{Z}$? A finite group action? $\endgroup$ – Uri Bader Feb 24 '17 at 19:53

There is (at least to the best of my knowledge) no direct equivalent of Kac's lemma for $\mathbb{Z}^d$ actions with $d>1$. In the paper of Aaronson and Weiss "A $\mathbb{Z}^d$ ergodic theorem with large normalising constants" there is a notion of Random Kac's inequality. This notion requires taking a Cartesian product of the action with a $\mathbb{Z}^d$ dyadic odometer.


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