Mapping equivalence under iterative dynamics for symbolic sequences

In order to employ powerful tools in signal processing to analyze symbolic sequences, a mapping is commonly applied first to transform the symbolic sequences to numerical sequences. Therefore it is important to investigate the role of the mapping in the final analysis results. The concepts of mapping equivalence and related theory have been proposed previously for the case where the data is processed by an operator for only once. However, many operators such as de-noising filter, smoothing filter and certain algorithm may be utilized multiple times. In this paper, we extend the concepts of mapping equivalence to the case of iterations of operators. We provide various theoretical results on determining the equivalence of two mappings. We also establish the connection of analysis robustness to the proposed mapping equivalence concepts. We provide numerical examples to illustrate and justify our theoretical results.