Automata, Borel functions and real numbers in Pisot base

UMR CNRS 6134, université de Corse, BP 52, 20250 CORTE, France; cagnard@univ-corse.fr; simonnet@univ-corse.fr

Abstract

This note is about functions ƒ : A^ω → B^ω whose graph is recognized by a Büchi finite automaton on the product alphabet A x B. These functions are Baire class 2 in the Baire hierarchy of Borel functions and it is decidable whether such function are continuous or not. In 1920 W. Sierpinski showed that a function is Baire class 1 if and only if both the overgraph and the undergraph of f are F_σ. We show that such characterization is also true for functions on infinite words if we replace the real ordering by the lexicographical ordering on B^ω. From this we deduce that it is decidable whether such function are of Baire class 1 or not. We extend this result to real functions definable by automata in Pisot base.