M. F. Barnsley. Iterated function systems for lossless data compression. In Fractals in Multimedia, M. F. Barnsley, D. Saupe, E. R. Vrscay (eds.), pp. 33-63, The IMA Volumes in Mathematics and Its Applications, Springer Verlag, 2002.
Iterated Function Systems with place-dependent probabilites are considered. Fascinating geometrical invariants, that apply even when there is no unique invariant measure, are presented. Furthermore, it is shown that the invariant measure of a stationary stochastic process, when it contains no atoms and is fully supported, can sometimes be associated with an IFS with probabilities, and with an associated dynamical system. This leads to the idea of an ergodic transform of a string of symbols, with respect to a given string; this is introduced and shown to be useful; it provides a unifying geometrical approach to the description of data compression algorithms such as Huffman, arithmetic, and the Burrows-Wheeler transform
@InCollection{Barnsley02,
Author = {Barnsley, M. F.},
Title = {Iterated function systems for lossless data compression},
BookTitle = {Fractals in Multimedia},
editor = {Barnsley, M. F. and Saupe, D. and Vrscay, E. R.},
Pages = {33--63},
Series = {The IMA Volumes in Mathematics and Its Applications},
Publisher = {Springer Verlag},
Year = {2002}
}