Čech cohomology of attractors of discrete dynamical systems
The question addressed in this paper is the following. Assume that \(K\) is an asymptotically stable attractor for a homeomorphism \(f:\mathbb R^n\to\mathbb R^n\), with basin of attraction \(A(K)\). What hypotheses are sufficient to ensure that the inclusion of \(K\) into \(A(K)\) induces an isomorphism of their Čech cohomology groups? The main results are that this holds if the coefficients of the cohomology groups are taken in \(\mathbb{Q}\) or \(\mathbb{Z}_p\), and that it holds for the integral Čech cohomology if and only if the cohomology of either \(K\) or \(A(K)\) is finitely generated. In addition, the cohomology of periodic point-free attractors of volume-contracting homeomorphisms of \(\mathbb{R}^3\) is computed.
scientific article; zbMATH DE number 6332804
scientific article