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A temporal central limit theorem for real-valued cocycles over rotations. (English. French summary) Zbl 1411.37007
Let \(0<\alpha<1\) be a badly approximable irrational number and \(0<\beta<1\) be badly approximable with respect to \(\alpha\), which implies the existence of some positive constant \(c\) such that \(|q\alpha -p|\ge c\) and \(|q\alpha -\beta -p|\ge c/|q|\) for any \(p\in\mathbb{Z},\ q\in\mathbb{Z}^*\) such that \(\mathrm{g.c.d.}(p,q)=1\).
Denote by \(R_\alpha\) the rotation by \(\alpha\) on \(\mathbb{T}= \mathbb{R}/\mathbb{Z}\), and for any \(x\in \mathbb{T}\) let \(f_\beta(x):= 1_{\{0\le x<\beta\}}-\beta\).
The authors consider the Birkhoff sums \(S_k(R_\alpha,f_\beta, x) := \sum_{j=0}^{k-1} f_\beta\circ R_\alpha^j(x)\), for \(k\in\mathbb{N}^*\).

The main result is the following temporal central limit theorem, generalizing a theorem by J. Beck [Period. Math. Hung. 60, No. 2, 137–242 (2010; Zbl 1259.11092); Period. Math. Hung. 62, No. 2, 127–246 (2011; Zbl 1260.11060)]: for any \(x\in \mathbb{T}\) there exist sequences \(B_n(\alpha,\beta)\) (not depending on \(x\)) and \(A_n(\alpha,\beta,x)\) such that for any \(a<b\), as \(n\to\infty\): \[ \frac{1}{n} \mathrm{Card}\left\{ k\in\{1,\ldots, n\}\Big| a < \frac{S_k(R_\alpha,f_\beta, x) - A_n(\alpha,\beta,x)}{B_n(\alpha,\beta)} < b\right\} \longrightarrow \frac{1}{\sqrt{2\pi}} \int_a^b e^{-t^2/2} dt. \]
The proof is based on a renormalization algorithm taking advantage of both the continued fraction expansion of \(\alpha\) and the Ostrowski expansion of \(\beta\), which provides a way of encoding the dynamics symbolically, in terms of a Markov chain. This symbolic coding allows to reduce the above main result to a central limit theorem for non-homogeneous Markov chains.

MSC:
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37A30 Ergodic theorems, spectral theory, Markov operators
37E10 Dynamical systems involving maps of the circle
11K06 General theory of distribution modulo \(1\)
11K38 Irregularities of distribution, discrepancy
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