Poisson stability for impulsive semidynamical systems (Q1044494)

A semidynamical system is a triple \((X, \pi, {\mathbb R}_+)\), where \(X\) is a metric space and \({\mathbb R}_+\) is the set of non-negative real numbers, and \(\pi: X\times{\mathbb R}_+\to X\) is a continuous map with \(\pi(x, 0)=x\) and \(\pi(\pi(x, t), s)=\pi(x, t+s)\), for all \(x\in X, t, s\in{\mathbb R}_+\). An impulsive semidynamic system \((X, \pi, M, I)\) consists of a semidynamic system, a nonempty closed subset \(M\) of \(X\) and a continuous function \(I: M\to X\). A subset \(U\subseteq X\) is positively \(\tilde{\pi}\)-recursive if for each \(T\geq0\) there is a time \(t>T\) and an element \(x\in V\) such that \(\tilde{\pi}(x, t)\in U\). A point \(x\in X\) is positively Poisson \(\tilde{\pi}\)-stable if every neighborhood of \(x\) is positively \(\tilde{\pi}\)-recursive with respect to \(\{x\}\). The paper gives some characterizations of the positively Poisson stable points. The paper proves that the following conditions are equivalent (1) \(x\) is positively Poisson stable (2) For any neighborhood \(U\) of \(x\) and \(T>0\), \(\tilde{\pi}(x, t)\in U\) for some \(t>T\); (3) \(x\in L^+(x)\); (4) \(\overline{\tilde{\pi}^+(x)}=L^+(x)\); (5) \(\tilde{\pi}^+(x)\subseteq L^+(x)\); (6) For all \(\varepsilon>0\), there exists \(t\geq1\) such that \(\tilde{\pi}(x, t)\in B(x, \varepsilon)\). The paper further proves that if \(\tilde{\pi}^+(x)=\cup\{\pi(x_j^+, [0,\phi(x_j^+)))\,|\,j=0,\dots, k\}\), \(\phi(x_k^+)<+\infty\), and \(\tilde{\pi}(x_j^+, [0,\phi(x_j^+)))\cap \tilde{\pi}(x_i^+, [0,\phi(x_i^+)))=\emptyset\) for \(i\not=j\), \(i, j=1, \dots, k\), the Poisson \(\tilde{\pi}\)-stable point \(x\) will be periodic. The paper also extends an earlier result by Nemyckii et al and proves that in the impulsive semidynamical system with \(X\) being a complete metric space, a positively Poisson \(\tilde{\pi}\)-stable but not eventually periodic point has the property \(\tilde{L}^+(x)-\tilde{\pi}^+(x)\) is dense in \(\tilde{L}^+(x)\). Moreover, the paper gives a condition for the set of positively Poisson \(\tilde{\pi}\)-stable points to be dense in \(X\), assuming all points in \(X\) is non-wandering which means every neighborhood of \(x\) is self-positively \(\tilde{\pi}\)-recursive.