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Critical values and representation of functions by means of compositions. (English) Zbl 0636.26004

If \(f: U\to R^ m\) is a continuously differentiable mapping defined on an open domain \(U\subset R^ n\), then \(\sum (f)=<x\in U: \| df(x)\| =0\}\) is a set of critical points of f and \(\Delta (f)=f(\sum (f))\) stands for a set of critical values of f. For a compact domain \(D\subset R^ n,\) \(C^ k(D,m)\) denotes the space of mappings: \(f: D\to R^ m,\) which can be extended to a k times continuously differentiable mapping of some neighbourhood of D. Let \(B^ i\subset R^{n_ i}\) be closed balls for \(i=1,...,s+1\) and let \(f: B^ 1\to B^{s+1}\) be given as a composition \(f=f\quad s\circ f^{s-1}\circ...\circ f^ 1,\) where \(f^ i: B^ i\to B^{i+1}\), \(f^ i\in C^{k_ i}(B^ i,n_{i+1}),\) \(k_ i\geq 2\). We call \(S=(n_ 1,k_ 1,n_ 2,k_ 2,...,n_ s,k_ s)\) the diagram of representation and say that f is representable with the diagram S. Let \(\sigma(S)=\sum^{s}_{i=1}(n_ i-n_{i+1})/(k_ i-1).\) The main result of the paper is the following Theorem: If f is representable with the diagram S, then \(\dim_ e\Delta (f)\leq \sigma (S),\) where \(\dim_ e\Delta(f)\) is the entropy dimension of \(\Delta(f)\). The paper is a continuation of an earlier work by the same author [Math. Ann. 264, 495-515 (1983; Zbl 0507.57019)].
Reviewer: W.Wilczyński

MSC:

26B40 Representation and superposition of functions
57R70 Critical points and critical submanifolds in differential topology
57R45 Singularities of differentiable mappings in differential topology
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory

Citations:

Zbl 0507.57019
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