 
Summary: Algebraic and Analytic Randomness
JeanPaul Allouche
CNRS, LRI, B“atiment 490
F91405 Orsay Cedex, France
allouche@lri.fr
http://www.lri.fr/¸allouche
Abstract. Is it possible to define mathematically words like: randomness, chaos,
disorder, irregularity, complexity, or like: determinism, order, periodicity, regularity,
simplicity? Are there concepts inbetween (quasiperiodicity)? How do these concepts
fit objects from physics, e.g., glasses, crystals, quasicrystals? We try to describe
and compare various notions used in mathematics.
1 A few ``principles''
In what follows we restrict our study to infinite sequences taking their values
in a finite set (sometimes called alphabet). To deal with more complicated
objects (e.g., functions) is possible by ``hacking'' them and restricting to the
``caricature'' of the object: if an object is random, all its caricatures should
also be random.
We begin with a few principles.
ffl No general definition of randomness is given in mathematics. Only defi
nitions suitable for a given purpose can be found.
