Conley index of isolated equilibria

Abstract

In this paper we study stable isolated invariant sets and show that the zeroth singular homology of the Conley index characterizes stability completely. Furthermore, we investigate isolated mountain pass points of gradient-like semiflows introduced by Hofer in \cite{4} and show that the first singular homology characterizes them completely. The result of the last section shows that for reaction-diffusion equations $$ \align u_{t}-\Delta u& = f(u),\\ u_{|\partial\Omega}& = 0, \endalign $$ the Conley index of isolated mountain pass points is equal to $\Sigma^{1}$ - the pointed $1$-sphere. Finally we generalize the result of {\cite{1, Proposition 3.3}} about mountain pass points to Alexander-Spanier cohomology.