Topological classification of one-dimensional attractors and repellers of \(A\)-diffeomorphisms of surfaces by means of automorphisms of fundamental groups of supports
type: Item
instance of: scholarly article
title: Topological classification of one-dimensional attractors and repellers of \(A\)-diffeomorphisms of surfaces by means of automorphisms of fundamental groups of supports
zbMATH Open document ID: 0940.37019
DOI: 10.1007/BF02169053
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-12-12T00:00:00Z
review text: \(A\)-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.
reviewed by: S. V. Zelik
Mathematics Subject Classification ID: 37G30
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 37E35
zbMATH DE Number: 1376347
zbMATH Keywords: \(A\)-diffeomorphisms
zbMATH Keywords: base sets
zbMATH Keywords: one-dimensional attractors
zbMATH Keywords: orientable manifold
zbMATH Keywords: non-wandering points
zbMATH Keywords: Lobachevskii plane
author: V. Z. Grines
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1376347