text: In order to create a coherent pattern, a material surface  should exert a sustained and consistent action on nearby trajectories throughout the time interval . Examples of such action are attraction, repulsion, or shear. In principle, any well-defined mathematical property qualifies that creates coherent patterns out of randomly selected nearby initial conditions.
  Most such properties can be expressed by strict inequalities. For instance, we call a material surface  attracting over the interval  if all small enough initial perturbations to  are carried by the flow into even smaller final perturbations to . In classical dynamical systems theory, invariant manifolds satisfying such an attraction property over infinite times are called attractors. They are not only special, but even locally unique in the phase space: no continuous family of attractors may exist.
  In contrast, in dynamical systems defined over a finite time interval , strict inequalities do not define exceptional (i.e., locally unique) material surfaces. This follows from the continuity of the flow map  over . For instance, if a material surface  attracts all nearby trajectories over the time interval , then so will any sufficiently close other material surface.
  Thus, attracting, repelling and shearing material surfaces are necessarily stacked on each other, i.e., occur in continuous families. This leads to the idea of seeking LCSs in finite-time dynamical systems as exceptional material surfaces that exhibit a coherence-inducing property more strongly than any of the neighboring material surfaces. Such LCSs, defined as extrema (or more generally, stationary surfaces) for a finite-time coherence property, will indeed serve as observed centerpieces of trajectory patterns. Examples of attracting, repelling and shearing LCSs are in a direct numerical simulation of 2D turbulence are shown in Fig.2a.