On invariant hypersurfaces of strongly monotone maps
Let \(f:\mathbb{R}^m \to \mathbb{R}^m\) be a \(C^k\), \(k\geq 1\), diffeomorphism. Let \(f\) have positive derivative, that is the Jacobian matrix has positive entries for any \(x\in \mathbb{R}^m\). \(x<y\) means natural partial ordering in \(\mathbb{R}^m\). We call a set \(A\subset \mathbb{R}^m\) unordered if no two of its points are related by \(<\). An invariant unordered topological hypersurface is called a nonmonotone manifold. Let \(T\) be an \(m\times m\)-matrix with positive entries, \(\Psi (T)= \min_{i,j,k,l} {T_{ik} T_{kl} \over T_{jk} T_{il}}\), \(\tau_B (T)= {1- \sqrt {\Psi(T)} \over 1+ {\Psi (T)}}\). Let \(K\subset \mathbb{R}^m\) be a compact set, \(\tau_B (f,K): =\sum_{x\in K} \tau_B (D (f(x)))\). If \(K\) is an attractor for \(g,d\) is a distance, \(B(K,g): =\{x\in X: \lim_{n \to\infty} d(g^n (x), K) =0\}\); \(|Df |_k|: =\sup_{x\in K} |Df(x) |\). Let \(M(f,K)\) denote the closure of the supports of all \(f\)-invariant Borel probability measures with support in \(K\). One of the main results is Theorem 2.1. Let \(V\subset \mathbb{R}^m\) be a nonmonotone manifold. Let \(K\subset V\) be an attractor for \(f|_V\) and let \(M= M(f,K)\). Suppose that (i) \(f\) is \(C^{1+k, r}\) for \(k\in \mathbb{N}\), \(0\leq r<1\), (ii) \(\tau_B (f,M)^{1+k+r} |Df |_M |^{k+r} <1\). Then \(B(K,f |_V)\) is \(C^{1+k,r}\). There are applications of the results to cooperative vector fields, in particular to the study of equilibria.
scientific article; zbMATH DE number 1038936
scientific article