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Journal Articles Ergodic Theory and Dynamical Systems Year : 2010

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.

Dates and versions

hal-00331667 , version 1 (17-10-2008)

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Cite

Olivier Durieu, Dalibor Volny. On Sums of Indicator Functions in Dynamical Systems. Ergodic Theory and Dynamical Systems, 2010, 30 (5), pp.1419-1430. ⟨10.1017/S0143385709000637⟩. ⟨hal-00331667⟩
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