Rare Events, Escape Rates and Quasistationarity: Some Exact Formulae

Abstract

Abstract. We present a common framework to study decay and exchanges rates in a wide class of dynamical systems. Several applications, ranging form the metric theory of continuons fractions and the Shannon capacity of contrained systems to the decay rate of metastable states, are given.

Extremes and Recurrence in Dynamical Systems

“…In [146], which appeared shortly after [49] on arXiv, the authors build up on the work of [147] and eventually obtain the dichotomy for balls and for conformal repellers. Then, in [141], making use of powerful spectral theory tools developed in [148], the dichotomy for balls is established for general systems such as those for which there exists as spectral gap for their respective Perron-Frobenius operator. In [142], the dichotomy for balls is obtained once again for the same type of systems considered in [141] but using as assumption the existence of decay of correlations against L 1 observables (see definition below).…”

“…Remember that we are under the assumption that m (µ ε × θ N ε )(φ > u m ) = m µ ε (φ > u m ) = m µ ε (U m ) → τ , when m → ∞; moreover it follows from the theory of [148] that h ε dν ε,m → h ε dLeb = 1, as m goes to infinity. In conclusion we get (1)) in the limit, as m → ∞.…”

A closing scheme for finding almost-invariant sets in open dynamical systems

“…Our focus is on systems with some (piecewise) smoothness and their ergodic properties. In recent years, there has been a burgeoning interest in the mathematical characterisation of these types of open systems [36,17,34,32,7]; see also the survey article [18] and references therein. Early work on the ergodic theory in one-dimensional dynamics was carried out by Pianigiani and Yorke [39], and further results have been established for a range of one-dimensional maps [9,10,11,6].…”

“…The similarity between the two characterisations has been exploited in [34,7], both of which treated metastable systems as perturbations of nonergodic systems, and in [32], which uses an approach, originally created to identify almost-invariant sets, to partition closed systems into two open systems with slow escape rates. The distinguishing feature of open systems is that the trajectories cannot re-enter the subset X r once they have fallen through the hole, whereas in metastable systems, trajectories can exit and reenter X 1 or X 2 with some small probability.…”

Behaviour of the escape rate function in hyperbolic dynamical systems

“…Also, although we usually suppress the parameters, ρ depends on both the hole, H, and on the measure, µ. In the present work, µ will always be either Lebesgue measure or the Sinai, Ruelle, Bowen (SRB) measure for f , and we study the behavior of the escape rate as a function of the hole.Many works on systems with holes have focused on the existence of quasi-invariant measures with physical properties by assuming either the existence of a finite Markov partition The derivative of the escape rate in the zero-hole limit has also received attention recently [BY,KL2,FP].In this paper we consider both small and large holes and make no Markovian assumptions on the dynamics of the open systems. Let H t be a 1-parameter family of holes varying continuously with t. Let ρ(t) = ρ(H t , µ), when defined.…”