# On Sums of Indicator Functions in Dynamical Systems

Abstract : In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exist a measurable set $A$ such that the sequence of the partial sums $\frac{1}{a_n}\sum_{i=0}^{n-1}(\ind_A-\mu(A))\circ T^i$ is dense in the set of the probability measures on $\R$. Further, in the ergodic case, we prove that there exists a dense $G_\delta$ of such sets.
Type de document :
Article dans une revue
Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2010, 30 (5), pp.1419-1430. 〈10.1017/S0143385709000637〉
Domaine :

https://hal.archives-ouvertes.fr/hal-00331667
Contributeur : Olivier Durieu <>
Soumis le : vendredi 17 octobre 2008 - 11:43:08
Dernière modification le : mardi 5 juin 2018 - 10:14:09

### Citation

Olivier Durieu, Dalibor Volny. On Sums of Indicator Functions in Dynamical Systems. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2010, 30 (5), pp.1419-1430. 〈10.1017/S0143385709000637〉. 〈hal-00331667〉

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