# 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

chaotic systems.

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.

- Last modified: 20/03/2019