Notes on a study of vector bundle dynamical systems. II: 1
Let \(K\) denote a locally trivial vector bundle with base space \(W\) (compact metrizable space) and with fiber \(E_x\) over \(x\in W\) \((n\)-dimensional real vector space). By the bundle dynamical system (or skew-product flow) is meant the fiber preserving one-parameter transformation group on \(K\), \[ \Phi_t: K\to K, \quad t\in\mathbb{R}. \tag{1} \] If, in addition, each \(\Phi_t\) maps fibers linearly onto fibers, we say that the bundle dynamical system is linear. For such linear system (1), the author defines so-called \(\Phi_t\)-admissible perturbation of the linear system (1) and studies some persistent perturbation properties, namely, the so-called pointwise boundedness property between orbits. More precisely, let \(\Psi_t\) be a perturbation of \(\Phi_t\) and let \(\overline u\) and \(u\) belong to the same fiber of \(K\). We say that the \(\Phi_t\)-orbit through \(\overline u\) is pointwise bounded from the \(\Psi_t\)-orbit through \(u\), if there is an increasing two-sided sequence \(\{t(j),\;j\in \mathbb{Z}\}\), \(t(0)=0\) and \(\lim_{j\to \pm \infty} t(j)= \pm\infty\), such that \(\sup_{j\in\mathbb{Z}} |\Phi_{t(j)} (\overline u)-\Psi_{t(j)} (u) |<\infty\).
scientific article; zbMATH DE number 1063988
scientific article