Large deviations in expanding random dynamical systems
The authors prove large deviations results for random dynamical systems [see also \textit{Y. Kifer}, in: Lyapunov exponents. Lect. Notes Math. 1486, 178-186 (1991; Zbl 0744.58061)]. Large deviations results in ergodic theory deal with occupational measures \(\mu_{x}^{n}=\frac{1}{n}\sum_{i=0}^{n-1}\delta_{\varphi^{i} x}\) on a topological dynamical system \((X,{\mathcal B},\mu,\varphi)\). Here \(X\) is a compact metric space, \({\mathcal B}\) is the Borel-\(\sigma\)-algebra on \(X\), \(\mu\) is a probability on \((X,{\mathcal B})\) and \(\varphi\) is a \(\mu\)-preserving continuous transformation on \(X\), the set of probability measures being topologized by weak convergence. For random dynamical systems the iteration of one single transformation is replaced by the successive application of randomly chosen transformations [see the thermodynamic formalism of \textit{T. Bogenschütz}, Random Comput. Dyn. 1, No. 1, 99-116 (1993; Zbl 0848.28010)]. In Kifer's work the results are concerned with occupational measures of the skew-product transformation induced by the random dynamical system. In the present paper the authors give a uniform rate function for large deviations for occupational measures of the corresponding cocycle.
scientific article; zbMATH DE number 1492606