Powers and iteration processes on modules. (English) Zbl 0821.39003
Given a left module \(V\) over a not necessarily commutative field \(K\) the authors introduce the structure of the so-called symmetric groupoid on a subset of \(V\). Then they extend the natural notion of nonnegative powers \(x^{\langle k \rangle}\) of the elements \(x\) of this groupoid to the whole \(V\). Assuming additionally that \(V\) is normed and \(K\) is valuated they study iteration processes on \(V\) of the type \(x \mapsto x^{\langle k \rangle} + c\) and analogons of the Mandelbrot set appearing here in a natural way. A special case of complex normed linear spaces is also described.
Reviewer: W.Jarczyk (Katowice)
MSC:
39B12 | Iteration theory, iterative and composite equations |
39B52 | Functional equations for functions with more general domains and/or ranges |
16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |
20N02 | Sets with a single binary operation (groupoids) |
28A80 | Fractals |
26A18 | Iteration of real functions in one variable |