Large deviation for return times

Abstract

We prove a large deviation result for return times of the orbits of a dynamical system in a r-neighbourhood of an initial point x. Our result may be seen as a differentiable version of the work by Jain and Bansal who considered the return time of a stationary and ergodic process defined in a space of infinite sequences.

Generalized dimensions, large deviations and the distribution of rare events

“…In this paper we have explored the relations between the spectrum of generalized dimensions D q and the recurrence properties of the dynamics. In fact, the former determines the large deviations of dynamical quantities such as return times [29] and hitting times: [21] and Proposition 2 herein. The statistics of hitting times ruled by Proposition 1 also opens the way to new techniques to estimate generalized dimensions via recurrence properties.…”

“…x = z. The first return time enjoys exponential large deviations, namely it was proven in [29] that:…”

“…The rate function Q * is slightly different from the Q given above. For its precise definition we refer again to [29] Theorem 2.5, where the large deviation property is proven 2 . The question has been asked whether a multifractal description of the first return time could be meaningful, by considering the set of points where the limit in Eq.…”

“…(28), with J 1 and J 2 defined in Eqs. (29) and (31), respectively. For the latter integral the inequality (31)…”

Extreme value distributions of observation recurrences

“…The latter manifest themselves on small, but not negligible, scales, a regime which we called penultimate [16]. The presence of large deviations for the point-wise dimensions has been rigorously proved for conformal repellers 13 in the paper [19]. Suppose we have an exact dimensional measure µ, call D 1 the µ-almost sure limit, and suppose that the following limit exists (55) D…”