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Processing of biological data sequences by mapping into numerical signals is a commonly used technique. The operators such as de-noising filter, smoothing filter and certain algorithm could be used iteratively. Little is known about the consistency of analysis results with different mapping strategies in this situation. Meanwhile, due to the errors and noises in acquisition of data, the stability of analysis results should never be neglected. In this paper, we provide a method for analyzing the consistency between different mappings under iterations of operator. We define different concepts of mapping equivalence. We show the necessary and sufficient condition for consistency under iteration of affine operator. We present a few theoretical results on the equivalent mappings on the concept of Fatou and Julia Set. We give the definition of stability under iteration of operator and show the stability issue can be viewed as a special case of mapping equivalence. We also establish the connection of stability to Fatou and Julia set. Finally, we present experimental results on human gene AD169 sequence and rhodopsin gene sequence under one of the widely used mappings and illustrate the equivalent mapping for a smoothing filter.