A classification of closedness of time scales under translations

The study of almost periodic and almost automorphic dynamic equations on time scales reveals some interesting dynamical behavior. Before studying almost periodicity and almost automorphy of dynamical behavior of solutions for dynamic equations, the first and important thing is to introduce and study almost periodic and almost automorphic functions on time scales. Since the closedness of time scales under translations plays a very significant role in establishing translation function theory, it is necessary to study the translation closedness of time scales and establish a theory of translation closedness of an arbitrary time scale. In this chapter, a classification of closedness of time scales under translations is introduced including CCTS, ACCTS and changing-periodic time scales. Moreover, the concept of time scale space is introduced and studied, which provides a new classification method of time scales with translation approximation property through almost periodic and almost automorphic function theory on the real line.