A computation method of Hausdorff distance for translation time scales

In this paper, we propose a computation method of Hausdorff distance between an arbitrary time scale and its translation (i.e. the endpoints approximation method or the EA method) and construct an explicit distance function which can be applied to calculate the Hausdorff distance. Furthermore, we construct a continuous linear broken line function with δ-accuracy to obtain the Hausdorff distance and its corresponding error intervals. Based on the linear construction of the distance function with δ-accuracy, the completeness of the distance function spaces is proved, then we introduce the concept of f-equivalent classes of time scales and embed the time scale spaces into the complete distance function spaces and some embedding theorems of time scales are established. Considering almost periodicity and almost automorphy of the distance functions on (Formula presented.), we study almost periodic time scales and introduce the concept of almost automorphic time scales. Moreover, some new properties and criteria of almost periodic and almost automorphic time scales are established and some examples are provided.