  Topological entropy is a common measure of the rate of mixing in a flow. It
can be computed by partition methods, or by estimating the growth rate of
material lines or other material elements. This requires detailed knowledge of
the velocity field, which is not always available, such as when we only know a
few particle trajectories (ocean float data, for example). We propose an
alternative approximation to topological entropy,applicable to two-dimensional
flows, which uses only a finite number of trajectories as input data. To
represent these sparse data sets, we use braids, algebraic objects that record
how strands, i.e., trajectories, exchange positions with respect to a
projection axis. Material curves advected by the flow are represented as
simplified loop coordinates. The exponential rate at which a braid deforms
loops over a finite time interval as the strands exchange places is the
Finite-Time Braiding Exponent (FTBE) and serves as a proxy for topological
entropy of the two-dimensional flow. We demonstrate that FTBEs are robust with
respect to the value of numerical time step,details of braid representation,
and choice of initial conditions inside the mixing region. We also explore how
closely the FTBEs approximate topological entropy depending on the number and
length of trajectories used.

Chaotic Dynamics;Dynamical Systems