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Limit sets and prolongations in pseudo-processes. (English) Zbl 0675.54038

This paper is concerned with the behaviour of limit sets and prolongations and stability for pseudo-processes. (X,G,H,\(\mu)\) is a pseudo-process if G is an abelian semigroup, H is a subsemi-group containing the neutral element 0, X is nonempty set and \(\mu\) maps \(G\times H\times H\) into X such that \(\mu (g,x,0)=x\) (g\(\in G\), \(x\in X)\) and \(\mu (g+h,\mu (g,x,h),k)=\mu (g,x,h+k)\) (g\(\in G\), \(x\in X\), \(h,k\in H)\). The author considers the case G being the reals, H the non-negative reals, and X a compact metric space. Pseudo-processes are a generalisation of abstract dynamical systems.
The author defines various stability-like conditions for pseudo-processes and defines positive “orbits” and limit sets in a natural way. It is shown, for example, that limit sets are closed and connected (under appropriate assumptions). The main theorems relate the stability conditions to the upper semi-continuity of (point of set) maps from points to their appropriate limit sets. These ideas and results are then extended to prolongations.
Reviewer: M.Sears

MSC:

54H20 Topological dynamics (MSC2010)
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