Through a continual process of stretching and folding, much like in a "baker's map,"
tracers advected in chaotic flows will develop into complex fractals.
The fractal dimension of a single contour will be between 1 and 2.
Exponential growth ensures that the contour, in the limit of very
long time integration, becomes fractal.
Fractals composed of a single curve are infinitely long and when
formed iteratively, have an exponential growth rate, just like an
advected contour.
The Koch Snowflake, for instance, grows at a rate of 4/3 per iteration.
The figure below shows the fractal dimension of an advected contour as a function
of time, measured in four different ways.  A good method of measuring the
fractal dimension of an advected contour is the uncertainty exponent.