Compact automata

Abstract

If now A is a compact metric set and if Φ0, Φ1,...,Φ q-1 are q continuous maps from A into A (q ≥ 2), it is easy to define the automatic action of the family (Φ j ) on A, and to exhibit, in this way, automatic sequences taking their values in the compact set A. Such an example is given by the sequence (e 2iπαSq(n)), where Sq(n), as usual, denotes the sum of digits in the q-adic expansion of n, andwhere α is any irrational number. © 2010 Springer-Verlag Berlin Heidelberg.