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MORPHISMS OF SEMI–DYNAMICAL SYSTEMS ON ALGEBRAIC STRUCTURES

Introduction. The notion of semi–dynamical system arose from the study of ordinary differential equations and was successively generalised by several authors [12], [1]. BUCUR–BEZZARGA [4] seem to be the first to accept the ramification points. In this way, some specific notions from potential theory were introduced within the framework of semi–dynamical systems (for example, excessive functions). My papers [14], [15], [16], continued the research in this direction, introducing and studying further notions, such as: entrance and exit boundaries, harmonic functions and potentials, mor-phisms, Green function, time changes and so on. Even after these developments , it seems a surprising result that this theory, with a purely determi-nistic character (at least at first sight), by its starting point, includes all the situations which appear in the axiomatic potential theory (hence essentially probabilistic). I defined the notion of semi–dynamical system on an algebraic structure (s.d.s.a.s.) (X, Φ), imposing that the set X possess a structure of ordered convex cone, and the map Φ : [0, +∞) × X → X has some natural properties of compatibility with the algebraic structure. A good duality theory can be developed (this was not available within the theory of general semi–dynamical systems, see [5]), each such system possessing a canonical dual of the same type. An abstract version for the potential theory associated with a kernel or with a resolvent family of kernels, is proposed in [8]. The definition of s.d.s.a.s. develops the analogous abstract theory for the semi–groups of kernels. It was showed that the following situations are subsumed to this notion: the semi–groups of kernels (on measurable spaces) [9], as well as of linear, positive operators (on linear ordered spaces). Particularly , with each semi–dynamical system one associates, using this procedure,