Large deviations for short recurrence (Q934232)

For a \(\psi\)-mixing dynamical system consider the function \(\tau(C_{n})/n, n=1,2,\dots\), where \(\tau(C_{n})\) is the first return time of a cylinder \(C_{n}\) to itself. \textit{B. Saussol, S. Troubetzkoy} and \textit{S. Vaienti} [J. Stat. Phys. 106, No. 3--4, 623--634 (2002; Zbl 1138.37300)] proved that \(\lim_{n\rightarrow\infty}\tau(C_{n})/n\) exists and is constant a.e. if the \(C_{n}\) are chosen in a descending sequence of cylinders around a given point. The present authors prove upper and lower bounds for the corresponding large deviation function. Actually, under mild assumptions they compute the large deviation function directly. The free energy function of \(\tau(C_{n})/n\) is also computed. Several examples illustrate the results obtained.