Topological classification of one-dimensional attractors and repellers of \(A\)-diffeomorphisms of surfaces by means of automorphisms of fundamental groups of supports
\(A\)-diffeomorphisms \(f\) defined on a two-dimensional closed smooth orientable manifold \(M\) of genus \(g\geq 0\) are considered. It is known that the set of non-wandering points of \(A\)-diffeomorphisms can be represented as a finite union of non-intersecting closed invariant sets, called base sets, each of them contains a dense orbit. It is also known that if a base set is one-dimensional then it is an attractor or a repeller. The problem of topological classification of \(A\)-diffeomorphisms restricted to their one-dimensional base sets is studied. The algebraic representation \((F,\tau_i)\), where \(F\) is a discrete subgroup of motions of the Lobachevskii plane and \(\tau_i\) is a hyperbolic automorphism of \(F\) is constructed for every \(C\)-dense component \(\Omega_i\) of the one-dimensional base set \(\Omega\). Moreover, it is proved that the diffeomorphisms \(f\big|_{\Omega}\) and \(f'\big|_{\Omega'}\), where \(\Omega\) and \(\Omega'\) are the corresponding one-dimensional base sets, are topologically conjugated if and only if the corresponding algebraic representations are algebraically conjugated.
scientific article; zbMATH DE number 1376347
scientific article