Approximating invariant densities of metastable systems

Abstract

We consider a piecewise smooth expanding map on an interval which has two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs. The result is generalized to the case of finitely many invariant components.

Almost sure invariance principle for sequential and non-stationary dynamical systems

“…We note that Assumption (A2), more precisely the fact that β −1 0 is strictly bigger than 2 instead of 1, is sufficient to get the uniform Doeblin-Fortet-Lasota-Yorke inequality (7.9) below, as explained in Section 4.2 of [17]. We now construct the family F by choosing maps T ε ∈ F close to T ε=0 := T in the following way:…”

On transfer operators and maps with random holes

“…Statistical aspects of such open systems have been addressed by many authors, see for instance [2,11,13]. Open dynamical systems have found interesting applications in physics [2,18,29], and more recently, after the pioneering work of [8,23], it was found that open systems are intimately related to studying metastable dynamical systems [3,4,5,12,14,17,20] and their applications in geophysical sciences [9,16].…”

Pseudo-orbits, stationary measures and metastability

“…Moreover, we use our result to identify random perturbations that exhibit a metastable behavior. Such a phenomenon has recently been a very active topic of research in both ergodic theory [3,4,14,16,17,24] and applied dynamical systems [11,30]. Section 2 contains the setup of the problem, our assumptions, the notion of a least element and the statement of our main result (Theorem 2.8).…”