Combinatorics of one-dimensional hyperbolic attractors of diffeomorphisms of surfaces.

*(English. Russian original)*Zbl 1079.37036
Dynamical systems and related problems of geometry. Collected papers dedicated to the memory of Academician Andrei Andreevich Bolibrukh. Transl. from the Russian. Moscow: Maik Nauka/Interperiodika. Proceedings of the Steklov Institute of Mathematics 244, 132-200 (2004); translation from Tr. Mat. Inst. Steklova 244, 143-215 (2004).

This paper is a survey and a generalization of the author’s results, dealing with a classification of one-dimensional hyperbolic attractors of diffeomorphisms of surfaces. This is achieved by solving a problem of topological conjugacy using an original combinatorial method permitting to describe hyperbolic attractors of such diffeomorphisms. The general framework is that of the theory of dynamical systems with hyperbolic attractors. In the past, the first results were due to R. F. Williams [Topology 6, 473–487 (1967; Zbl 0159.53702) and Publ. Math., Inst. Hautes Étud. Sci. 43, 169–203 (1973; Zbl 0279.58013)] (conditions for attractor topological conjugacy in terms of generalized solenoids, but without efficient procedure for their verification), followed by those of Plykin, Grines, Kalaï giving a description and a classification of the attractor topological structure, but here also without a constructive method for verifying conjugacy conditions. The paper’s author gives the conjugacy conditions that admit such verification.

The author’s approach permits to improve the problem statement by including the enumeration problem for attractors. Such a problem is related to the description of the set of all these attractors, the “complexity” of which does not exceed a given value. Here, naturally the topological entropy is taken as degree of complexity. Using a combinatorial method, the author essentially solves two problems. The first one, given two one-dimensional attractors of two diffeomorphisms \(f\) and \(g\) related to two closed surfaces \(M\) and \(N\), orientable or not, does there exist a homeomorphism \(h\) of certain neighborhoods of attractors leading to their topological conjugacy? Then the second one, given \(h>0\), find a representative of each class of topological conjugacy of attractors, with a given structure of accessible boundary, such that the topological entropy is not larger than \(h\) (problem of attractor enumeration).

For the entire collection see [Zbl 1064.37002].

The author’s approach permits to improve the problem statement by including the enumeration problem for attractors. Such a problem is related to the description of the set of all these attractors, the “complexity” of which does not exceed a given value. Here, naturally the topological entropy is taken as degree of complexity. Using a combinatorial method, the author essentially solves two problems. The first one, given two one-dimensional attractors of two diffeomorphisms \(f\) and \(g\) related to two closed surfaces \(M\) and \(N\), orientable or not, does there exist a homeomorphism \(h\) of certain neighborhoods of attractors leading to their topological conjugacy? Then the second one, given \(h>0\), find a representative of each class of topological conjugacy of attractors, with a given structure of accessible boundary, such that the topological entropy is not larger than \(h\) (problem of attractor enumeration).

For the entire collection see [Zbl 1064.37002].

Reviewer: Christian Mira (Quint)

##### MSC:

37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |

37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |

37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |

37B40 | Topological entropy |

37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |

##### Keywords:

surface diffeomorphisms; smooth dynamical systems; hyperbolic attractors; topological entropy; conjugacy of attractors
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\textit{A. Yu. Zhirov}, in: Dynamical systems and related problems of geometry. Collected papers dedicated to the memory of Academician Andrei Andreevich Bolibrukh. Transl. from the Russian. Moscow: Maik Nauka/Interperiodika. 132--200 (2004; Zbl 1079.37036); translation from Tr. Mat. Inst. Steklova 244, 143--215 (2004)

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