Automata, Borel functions and real numbers in Pisot base

This note is about functions f : A ! ! B ! whose graph is recognized by a Buchi finite automaton on the product alphabet A◊B. These functions are Baire class 2 in the Baire hierarchy of Borel functions and it is decidable whether such fonction are continuous or not. In 1920 W. Sierpinski showed that a function f : R ! R is Baire class 1 if and only if both the overgraph and the undergraph of f are Fae. 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 ! . From this we deduce that it is decidable whether such fonction are of Baire class 1 or not. We extend this result to reals functions definable by automata in Pisot base.

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