Large deviations, fluctuations and shrinking intervals
The purpose of this paper is to model how entropy may increase or decrease over time before it reaches an equilibrium. Let \(T:\Lambda\to\Lambda\) be a mixing hyperbolic diffeomorphism, \(\mu\) an equilibrium state for a Hölder continuous function and \(\Psi: \Lambda\to\mathbb{R}\) a Hölder continuous function with \(\int\Psi d\mu> 0\). Let \({\mathcal M}_T\) be the set of \(T\)-invariant Borel probabilty measures and \({\mathcal I}_\Psi= \{\int\Psi d\mu: m\in{\mathcal M}_T\}\) (non-trivial unless \(\Psi\) corresponds to the measure of maximal entropy). Set \(\Psi^n(x)= \sum^{n-1}_{k=0}\Psi(T^k x)\). The fluctuation theorem says that if \(\mu_\Psi\) is not the measure of maximal entropy, then \(\int\Psi d\mu> 0\) and there exists \(p^*> 0\) such that if \(|p|< p^*\) then \[ \lim_{\delta\to 0}\,\lim_{n\to\infty}\,{1\over n}\log\Biggl({\mu\{x: {1\over n}\Psi^n(x)\in (p-\delta, p+\delta)\}\over \mu\{x: {1\over n}\Psi^n(x)\in (-p+\delta, -p+\delta)\}}\Biggr)= p. \] Suppose that \(\delta_n\) decreases to zero sufficiently slowly. The authors ask: does there exist an analytic rate function \(I:\text{Int}({\mathcal I}_\Psi)\to \mathbb{R}^+\) with \(I(p)= 0\) if and only if \(p= \int\Psi d\mu\) such that \[ \lim_{n\to\infty}\,{1\over n}\log\mu_\Phi\Biggr(x: {\Psi^n(x)\over n}\in (p-\delta_n, p+\delta_n)\Biggr)= -I(p)? \] They show that the answer is always affirmative given modest conditions on \(\Psi\), if \(\delta^{-1}_n\) grows no faster than \(n^{1+\kappa}\), for some \(\kappa= \kappa(\Psi)> 0\). If \(T\) has a time reversal symmetry: there exists an involution \(i: \Lambda\to\Lambda\) such that \(i\circ T\circ i= T^{-1}\), then a corresponding version of the fluctuation theorem is obtained. This work is a continuation of work of \textit{Y. Kifer} [Trans. Am. Math. Soc. 321, No. 2, 505--524 (1990; Zbl 0714.60019)] and \textit{S. Orey} and \textit{S. Pelikan} [Trans. Am. Math. Soc. 315, No. 2, 741--753 (1989; Zbl 0691.60022)].
scientific article; zbMATH DE number 5655462
scientific article