A semi-invertible Oseledets Theorem with applications to transfer operator cocycles

A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems

“…In applications, transfer operator based methods were initially investigated in the area of molecular dynamics [10], and later in the context of geophysical flows, starting with the work of Froyland, Padberg, England, and Treguier [16]. This approach was later developed to identify and track time-varying structures, beginning with the works of Froyland, Lloyd and Quas on multiplicative ergodic theory in [12,13]. A survey of these techniques is provided in [21].…”

“…That is, for systems whose evolution rules change over time. This was initially developed by Oseledets in the 1960's [37] and was adapted and expanded to the semi-invertible setting in [12,13,22,23]. This extension is crucial to the study of transfer operators of non-autonomous dynamical systems because it covers cases where the dynamics are not necessarily invertible.…”

Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems

“…Having in mind applications to the detection of coherent structures in oceanic and atmospheric dynamics, the recent works of Froyland, Lloyd, Quas and González-Tokman [43,53,54] extend infinite dimensional versions of the multiplicative ergodic theorem to the setting of semi-invertible cocycles. Based on [91], Froyland, Lloyd and Quas established a multiplicative ergodic theorem in the context of Pcontinuous cocycles [43].…”

“…Having in mind applications to the detection of coherent structures in oceanic and atmospheric dynamics, the recent works of Froyland, Lloyd, Quas and González-Tokman [43,53,54] extend infinite dimensional versions of the multiplicative ergodic theorem to the setting of semi-invertible cocycles. Based on [91], Froyland, Lloyd and Quas established a multiplicative ergodic theorem in the context of Pcontinuous cocycles [43]. Building on [75] and [30], Quas and González-Tokman replaced the continuity condition by a strong measurability condition, together with the requirement that the Banach space X (and its dual X * ) is separable [53,54].…”

On the number of invariant measures for random expanding maps in higher dimensions

“…A difficulty of this setting is that there is no known formula for an averaged transfer operator that corresponds to the one described in (2) in [30,6]. The way we overcome this obstacle, as it has been done in [9,12,15], is by developing a random Lasota-Yorke inequality, Equation (9), which we use to prove several results in this paper.…”