The Conley index for discrete semidynamical systems
An improved homotopy index of isolated invariant sets of discrete semidynamical systems is introduced. It is based on the concept of so-called category of objects equipped with a morphism defined as follows. For a given category \(K\) let \(K_m\) denote a new category in which the objects are pairs \((X, f)\), where \(X\) is an object in \(K\) and \(f: X\to X\) is a morphism in \(K\). In order to define morphisms in \(K_m\), for objects \((X, f)\) and \((X', f')\) put \[ M((X, f), (X', f'))= \{(g, n): g: X\to X'\text{ is a morphism in }K, n\in \mathbb{Z}^+, g\circ f= f'\circ g\}. \] In \(M((X, f), (X', f'))\) introduce an equivalence relation \(\sim: (g_1, n_1)\sim (g_2, n_2)\) iff there exists \(k\in \mathbb{Z}^+\) such that \(g_1\circ f^{n_2+ k}= g_2\circ f^{n_1+ k}\). Morphisms \((X, f)\to (X', f')\) are defined as the equivalence classes of elements of \(M((X, f), (X', f'))\) in \(\sim\). By \(Htop\) denote the category in which objects are pointed topological spaces and morphisms from \(X\) to \(X'\) are the homotopy classes of continuous maps \(X\to X'\) which preserve the base point. Let \(S\) be an isolated invariant set for a continuous map \(\phi: Z\to Z\), where \(Z\) is a metric space (in practice: locally compact also). If \(N= (N_1, N_0)\) is an index pair (in the Mrozek's sense) for \(S\) then the index consists of the class of all objects isomorphic to \((N_1/N_0, \phi_N)\) in the category \(Htop_m\), where \(\phi_N: N_1/N_0\to N_1/N_0\) is given by \(\phi_N([x])= [\phi(x)]\) if \(x,\phi(x)\in N_1\backslash N_0\) and \(\phi_N([x])= [N_0]\) in the other case. The index defined in this way generalizes both the shape index of Robbin and Salamon, and the Mrozek's algebraic index based on the concept of Leray functor. A proof of the continuation property of the index and a construction of the corresponding connected simple system are also given in the reviewed paper.
scientific article; zbMATH DE number 823481