A Generalization of Formal Model of Internal Measurement: A Construction on One-Dimensional Maps

A generalization of formal model of internal measurement proposed by one of the authors (Gunji, et al. (1997). Physica D 110, 289-312) is addressed. An interface between defining a fixed point and holding an adjunction is constructed motivated by the problem of the gap between parts and wholeness in complex systems. We define a construction of the generalized procedure of formal model of internal measurement on families of one-dimensional maps. We introduce two-dimensional time evolutionary systems by examining computation of time evolutionary steps of one-dimensional maps in terms of a negotiation process between a collection of parts and wholeness. We discuss a power law behavior or fractals in some concrete one-dimensional maps.