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Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-preserving dynamical systems

  • *Corresponding author: Tanja I. Schindler

    *Corresponding author: Tanja I. Schindler

The authors are partially supported by the PRIN Grant 2017S35EHN "Regular and stochastic behaviour in dynamical systems" of the Italian Ministry of University and Research (MUR), Italy. This research is part of the authors' activity within the UMI Group "DinAmicI" www.dinamici.org and of the first author's activity within the Gruppo Nazionale di Fisica Matematica, INdAM, Italy. The second author acknowledges the support of the Centro di Ricerca Matematica Ennio de Giorgi and of UniCredit Bank R & D group for financial support through the 'Dynamics and Information Theory Institute' at the Scuola Normale Superiore

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  • We consider a conservative ergodic measure-preserving transformation $ T $ of a $ \sigma $-finite measure space $ (X, {\mathcal B},\mu) $ with $ \mu(X) = \infty $. Given an observable $ f:X\to \mathbb R $, we study the almost sure asymptotic behaviour of the Birkhoff sums $ S_Nf(x) : = \sum_{j = 1}^N\, (f\circ T^{j-1})(x) $. In infinite ergodic theory it is well known that the asymptotic behaviour of $ S_Nf(x) $ strongly depends on the point $ x\in X $, and if $ f\in L^1(X,\mu) $, then there exists no real valued sequence $ (b(N)) $ such that $ \lim_{N\to\infty} S_Nf(x)/b(N) = 1 $ almost surely. In this paper we show that for dynamical systems with strong mixing assumptions for the induced map on a finite measure set, there exists a sequence $ (\alpha(N)) $ and $ m\colon X\times \mathbb N\to \mathbb N $ such that for $ f\in L^1(X,\mu) $ we have $ \lim_{N\to\infty} S_{N+m(x,N)}f(x)/\alpha(N) = 1 $ for $ \mu $-a.e. $ x\in X $. Instead in the case $ f\not\in L^1(X,\mu) $ we give conditions on the induced observable such that there exists a sequence $ (G(N)) $ depending on $ f $, for which $ \lim_{N\to\infty} S_{N}f(x)/G(N) = 1 $ holds for $ \mu $-a.e. $ x\in X $.

    Mathematics Subject Classification: Primary: 37A40; Secondary: 37A25, 60F15.

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