Classical theory of continuous dynamical systems, defined for all time on a smooth phase manifold, attempts to classify the dynamical complexity of these systems by identifying asymptotic behavior of their solutions. The asymptotic procedures allow to extract the essential characteristics of such systems based on their behavior in the neighborhood of some lower-dimensional, flow-invariant objects (e.g hyperbolic/elliptic fixed points or periodic orbits, or more exotic recurrent sets). However, in the finite-time setting for non-autonomous dynamical systems, when the system is defined or known over a bounded time interval, the classical time-asymptotic notions of hyperbolicity or stability do not apply and new concepts and tools have to be developed in order to study the so called `Lagrangian transport' in such systems.

A major motivation for this problem arises from the desire to study transport and mixing problems in geophysical flows (e.g., spread of pollutants or phytoplankton) where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field.