Examples of one-dimensional hyperbolic attractors on nonorientable surfaces
type: Item
instance of: scholarly article
title: Examples of one-dimensional hyperbolic attractors on nonorientable surfaces
zbMATH Open document ID: 0969.37015
DOI: 10.1007/BF02675083
author: A. Yu. Zhirov
published in: Mathematical Notes
publication date: 2001-09-25T00:00:00Z
review text: This paper deals with some problems related to the existence of diffeomorphisms generating hyperbolic attractors on nonorientable surfaces. From the point of view of the ``realizability'' of an attractor on a surface, here attractors are studied up to topological conjugacy of their generating diffeomorphisms on the attractor neighborhoods. An order relation on the set of closed compact surfaces is introduced, for defining the minimal surface of the attractor of a diffeomorphism. The attractor is the union of the unstable manifolds of its points. The definitions of bunch, degree of a bunch, orientability of an attractor on surfaces are recalled. Noting that the attractor realization depends on its boundary type, and the orientability, the basic question arises as to what boundary types of orientable and nonorientable attractors, formally permissible by the Euler-Poincaré formula for a given surface, are actually realizable. The paper is devoted to the nonorientable case, which remainded open. Examples of diffeomorphisms with attractors permit to prove four theorems.
Mathematics Subject Classification ID: 37D05
Mathematics Subject Classification ID: 37C70
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 57R50
zbMATH DE Number: 1470982
zbMATH Keywords: maps
zbMATH Keywords: hyperbolic attractors
zbMATH Keywords: nonorientable surfaces
zbMATH Keywords: projective plane
zbMATH Keywords: topological conjugacy
zbMATH Keywords: topological entropy
reviewed by: Christian Mira
MaRDI profile type: Publication
cites work: Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms
description: scientific article; zbMATH DE number 1470982