On a structure of the fixed point set of homogeneous maps
For an integer \(m\geq 0\), a map \(f:\mathbb R^n\to \mathbb R^n\) is a polynomial \(m\)-homogeneous map if \(f(\mu x)=\mu^mf(x)\) for all \(x\in \mathbb R^n\), \(\mu\in \mathbb R\). The null set of \(f\) is denoted by \(\text{Nul}(f)\) and the set of all non-zero fixed points of \(f\) by \(\text{Fix}(f)\). Let \(V\) be an \(n\)-dimensional real vector space. Let \(\text{Sym}^m(V)\) denote the \(m\)-th symmetric tensor power of \(V\). The paper is concerned with four problems: (1) to find algebraic conditions on sets \(P\) and \(N\) in \(\mathbb C^n\), \(P\cap N=\emptyset\), \(0\in N\), such that there exists a polynomial \(m\)-homogeneous map \(f\) with \(P\subset \text{Fix}(f)\) and \(N\subset \text{Nul}(f)\); (2) in the case that such an \(f\) exists, to determine when it is unique; (3) to extend the sets \(P\) and \(N\) to the spectrum \(S_f:=\text{Fix}(f)\cup\text{Nul}(f)\) of \(f\) without giving and explicit construction of \(f\); (4) to study the topological and dynamical properties of a polynomial \(m\)-homogeneous map \(f\) determined by its spectrum \(S_f\).
scientific article; zbMATH DE number 6144761