Dynamics in Ultra-Pseudometric Spaces

Abstract

As we have already discussed in Chapter 3, dynamical systems over p-adic trees have a large number of usual and fuzzy cycles. This is one of the main disadvantages of the model of the process of thinking presented in Chapter 8: starting with an initial state x ₀ ∈ ℤ _p (or a ball B in the mental space) the brain of a cognitive system τ will often obtain no definite solution (no attractors!) However, as we shall see in this chapter, cycles of balls produce attractors in the space of ideas! Hence by developing the ability to work with collections of p-adic balls cognitive systems transferred cyclic-disadvantage into the great advantage: richness of cyclic behavior on the level of balls implies richness of the set of possible ideas-solutions. In this chapter we prove the existence of attractors in the space of collections of balls as well as present algorithms for finding these attractors, see [109] and [111].