Ergodic Theory and Dynamical Systems


Dynamical systems, shape theory and the Conley index

Joel W. Robbina1 and Dietmar Salamona2

a1 Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA

a2 Mathematics Institute, University of Warwick, Coventry CV47AL, UK


The Conley index of an isolated invariant set is defined only for flows; we construct an analogue called the ‘shape index’ for discrete dynamical systems. It is the shape of the one-point compactification of the unstable manifold of the isolated invariant set in a certain topology which we call its ‘intrinsic’ topology (to distinguish it from the ‘extrinsic’ topology which it inherits from the ambient space). Like the Conley index, it is invariant under continuation. A key point is the construction of a certain ‘index category’ associated with the isolated invariant set; this construction works equally well for flows or discrete time systems, and its properties imply the basic properties of both the Conley index and the shape index.