Local determinacy of abstract local dynamical systems

Abstract

O. Hajek proved in his book “Dynamical Systems in the Plane” (Chapter III) that there isat most one abstract local dynamical system which is locally equivalent to, or equivalently an extension of, a given elementary dynamical system, and suggested a question of finding reasonable conditions on the latter for the existence ofat least one such abstract local dynamical system. An elementary dynamical systemμ is said to satisfy the “No-Intersection Axiom” and is called an abstract germ ifμ(x1, t) = μ(x2,t) impliesx1 =x2. We show thatμ is (uniquely) extendable to an abstract local dynamical system if and only ifμ is an abstract germ, and hence the question is completely answered. After introducing various kinds of isomorphisms of abstract germs and abstract local dynamical systems corresponding to those of continuous germs and continuous local dynamical systems, we obtain some sufficient conditions for extendability of isomorphisms and possibility of restriction of them, and thus establish the local determinacy of abstract local dynamical systems up to isomorphisms in some wider categories.