Strongly Almost Periodic Sequences under Finite Automata Mappings

The notion of almost periodicity nontrivially generalizes the notion of periodicity. Strongly almost periodic sequences (=uniformly recurrent infinite words) first appeared in the field of symbolic dynamics, but then turned out to be interesting in connection with computer science. The paper studies the class of eventually strongly almost periodic sequences (i. e., becoming strongly almost periodic after deleting some prefix). We prove that the property of eventual strong almost periodicity is preserved under the mappings done by finite automata and finite transducers. The class of almost periodic sequences includes the class of eventually strongly almost periodic sequences. We prove this inclusion to be strict.