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 S _q (n), as usual, denotes the sum of digits in the q-adic expansion of n, andwhere α is any irrational number.