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Interval expanding maps and limit theorems. (Transformations dilatantes de l’intervalle et théorèmes limites.) (French) Zbl 0988.37032
Broise, Anne et al., Études spectrales d’opérateurs de transfert et applications. Paris: Société Mathématique de France, Astérisque. 238, 1-109 (1996).
From the text: A fine synthesis on interval expanding maps is made. The author proves the existence of invariant measures which are absolutely continuous with respect to Lebesgue measure for a larger class of expanding maps. Then she proves central and local limit theorems, gives the convergence rate and some conditions on the variance annihilation based on the periodic points of the map. The author makes precise these limit theorems by large deviation theorems. Finally she shows how to apply these theorems on various examples. This paper is an extension of the work of J. Rousseau-Egele [Ann. Probab. 11, 772-788 (1983; Zbl 0518.60033)].
In more detail, let \(f\) be a given function of bounded variation, denote \(S_nf= \sum_{k=0}^{n-1} f\circ T^k\) and \(\sigma^2= \lim_{n\to+\infty} \frac{1}{n} \int_0^1 (S_nf)^2h dm\geq 0\); if \(\sigma\) is different from zero, then the following is proved for \(S_n(f)\):
– a central limit theorem with convergence rate: \[ \sup_{v\in \mathbb{R}} \Biggl|hm \Biggl[ \Biggl\{ x\in [0,1]: \frac{S_nf(x)-nm(fh)} {\sigma\sqrt{n}}\leq v\Biggr\}\Biggr]- \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{t^2/2}e^{-t^2/2}dt \Biggr|\leq \frac{C}{\sqrt{n}}, \] – a local limit theorem: for each interval \(\Delta\) of \(\mathbb{R}\) of finite length uniform for \(z\in \mathbb{R}\), we have \[ \lim_{n\to+\infty} \biggl|\sigma\sqrt{n} hm \bigl[\{ x\in [0,1]: z+S_nf(x)- nm(fh)\in \Delta\}\bigr]- \frac{m(\Delta)}{2\pi} e^{\frac{-z^2}{2\sigma^2n}} \biggr|= 0. \]
For the entire collection see [Zbl 0893.00017].

MSC:
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
60F05 Central limit and other weak theorems
28D05 Measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37E05 Dynamical systems involving maps of the interval (piecewise continuous, continuous, smooth)
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