On-Line Odometers for Two-Sided Symbolic Dynamical Systems

We consider biinfinite sequences on a finite alphabet of digits which satisfy a constraint of finite type. Such sequences are perturbed by adding a 1 in position 0. The odometer is the function which transforms the initial sequence into an admissible sequence equivalent to the perturbed one. It is shown that the odometer can be realized by an on-line finite automaton when the constraint is linked to numeration in base β, where β is a Pisot number satisfying the equation βm = t1βm-1+...+tm, where t1 ≥ t2 = ... tm ≥ 1 are integers.

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