Finite-automaton transformations of strictly almost-periodic sequences

Different versions of the notion of almost-periodicity are natural generalizations of the notion of periodicity. The notion of strict almost-periodicity appeared in symbolic dynamics, but later proved to be fruitful in mathematical logic and the theory of algorithms as well. In the paper, a class of essentially almost-periodic sequences (i.e., strictly almost-periodic sequences with an arbitrary prefix added at the beginning) is considered. It is proved that the property of essential almost-periodicity is preserved under finite-automaton transformations, as well as under the action of finite transducers. The class of essentially almost-periodic sequences is contained in the class of almost-periodic sequences. It is proved that this inclusion is strict.