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The study of transport and mixing processes in dynamical systems isparticularly important for the analysis of mathematical models of physicalsystems. Barriers to transport, which mitigate mixing, are currently thesubject of intense study. In the autonomous setting, the use of transferoperators (Perron-Frobenius operators) to identify invariant andalmost-invariant sets has been particularly successful. In the nonautonomous(time-dependent) setting, \emph{coherent sets}, a time-parameterised family ofminimally dispersive sets, are a natural extension of almost-invariant sets.The present work introduces a new analytic transfer operator construction thatenables the calculation of \emph{finite-time} coherent sets (sets are thatminimally dispersive over a finite time interval). This new construction alsoelucidates the role of diffusion in the calculation and we show how propertiessuch as the spectral gap and the regularity of singular vectors scale withnoise amplitude. The construction can also be applied to general Markovprocesses on continuous state space.

An analytic framework for identifying finite-time coherent sets in time-dependent dynamical systems