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About the homological discrete Conley index of isolated invariant acyclic continua. (English) Zbl 1291.37021

Let \(U\subset\mathbb{R}^d\) be open, \(f:U\to f(U)\subset \mathbb{R}^d\) a homeomorphism, and let \(X\subset U\) be an isolated invariant set for the discrete dynamical system induced by \(f\). Suppose \(X\) is an acyclic continuum. Let \(h_*\) denote the Čech homology with rational coefficients. The authors prove that the trace of the first homological Conley index \(h_1(f,X)\) of \(f\) and \(X\) is at least \(-1\), and they describe the periodic behavior of trace\((h_1(f^n, X))\). If trace\((h_1(f,X))=-1\) then trace\((h_r(f,X))=0\) for each \(r>1\).
They also investigate the sequence of fixed point indices \(i(f^n,X)\). In case \(X=\{p\}\) is a fixed point and \(f\) is orientation-reversing, they characterize the possible sequences \((i(f^n,p))_{n\geq 1}\). As a corollary it is proved that there are no minimal orientation-reversing homeomorphisms in \(\mathbb{R}^3\).
The paper contains an exposition of the discrete Conley index and its duality.

MSC:

37B30 Index theory for dynamical systems, Morse-Conley indices
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
54H25 Fixed-point and coincidence theorems (topological aspects)
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