New statistical perspectives on chaotic attractors
It is well known that strange attractors are characterised
by their fractal dimensions, which quantify the mass
clustered into a small ball. Recent work, using statistical
approaches, has revealed other generic properties of
The fractal dimensions characterise the dense regions
of the attractor using a power-law, but the distribution of
density in the sparse regions is also characterised by a
power-law, which we term the 'lacunarity exponent'.
The fractal dimension describes the mass of the attractor
contained in small regions, but it is also possible to study the
shape of clusters of points which sample the attractor. The
statistics of the shape of these clusters is characterised by
power laws. The exponents of these lower-laws are found to
exhibit phase transitions.
Physical applications of these phenomena will also
be discussed, including particles advected in fluid flows
and ray trajectories in random media.