Characteristics of Singular Entities of Simple Cell Mappings
For a given simple cell mapping C(z) there is a cell mapping increment function F(z, C K ) associated with the mapping C K (z). In Chapter 5 we have considered the singular entities of such cell functions. These include non-degenerate and degenerate singular k-multiplets m k , k ∈ N + 1, and cores of singular multiplets. We recall here that for dynamical systems governed by ordinary differential equations (Coddington and Levinson , Arnold ) and for point mapping dynamical systems (Hsu , Bernussou ), the singular points of the vector fields governing the systems can be further classified according to their stability character. In this spirit one may wish to classify singular entities of cell functions according to their “stability” character and to see how they influence the local and global behavior of the cell mapping systems. On this question a special feature of cell mappings immediately stands out. Since a cell function maps an N-tuple of integers into an N-tuple of integers, the customary continuity and differentiability arguments of the classical analysis cannot be used, at least not directly. Evidently, a new framework is needed in order to delineate various mapping properties of the singular entities of cell functions.
KeywordsSingular Point Point Mapping Mapping Property Mapping Motion Rootedness Property
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