Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

I am not trying to avoid my 19-20 reference list, but it is remarkable
how many other references there are to distract one and yet remain on
topic!

A case in point is the volume Fractals Non-integral dimensions and
applications, G. Cherbit (U. Paris VII) (Ed.), Wiley: Chichester 1991,
originally published as Fractals: Dimensions non-entieres et
applications, Masson: Paris 1987.

One of the best papers in this volume is "Construction of fractals and
dimension problems" by F. M. Dekking (Mathematics Dept. U Delft,
Netherlands) who goes back to Peano's constructions published in 1890
that involve substitution of codes which are morphisms under
concatenation of words: f(WW' ) = f(W)f(W' ) beginning with the set S =
{e, n, w, s} and the set S* which is the union fo all nonnegative
integral k of S^k is mapped into itself beginning with:

1) e-->eneswsene
2) n-->nwnesenwn
3) w-->wswnenwsw
4) s-->seswnwses

More generally, for an aphabet S and a mapping f: S* --> R^d such that:

5) f(WW' ) = f(W) + f(W' )

which associates with each word W a point in R^d, a substitution theta:
S* --> S* is said to admit a representation in R^d if:

6) f(theta(s)) = L(f(s)), s in S

for some homomorphism f: S* --> R^d and some linear mapping L:
R^d-->R^d. L is called "expansive" if its proper values have modulus
greater than 1. Readers can see for example in one dimension that
f(theta(s)) = L(f(s)) > f(s) means that theta increases f(s).
This leads to fractals, although there are also other routes to
fractals.

Osher Doctorow

.

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