Abstract
Axiom A diffeomorphisms of closed 2-manifold of genus \(p \geqslant 2\) whose nonwandering set contains a perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to a pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of this paper is as follows. Two diffeomorphisms from the given class are topologically conjugate on perfect spaciously situated attractors if and only if the corresponding homotopic pseudo-Anosov homeomorphisms are topologically conjugate by means of a homeomorphism that maps a certain subset of one pseudo-Anosov homeomorphism onto a subset of the other.
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Notes
A basic set \(\Lambda \) of an\(A\)-diffeomorphism f is called an attractor if there exists a closed neighborhood U of \(\Lambda \) such that \(f(U) \subset {\text{int}}U\), \(\bigcap\limits_{j \geqslant 0} \,{{f}^{j}}(U) = \Lambda \). An attractor of the diffeomorphism \({{f}^{{ - 1}}}\) is called a repeller of the diffeomorphism \(f\).
\({{[x,y]}^{s}},{{[x,y]}^{s}},{{(x,y)}^{u}},{{(x,y)}^{u}}\) denote closed and open intervals bounded by the points \(x,y\) that are contained in the one-dimensional stable \(W_{x}^{s}\) and unstable \(W_{x}^{u}\) manifolds, respectively.
For a domain \(\Delta \), its boundary accessible from within is defined as a subset \(C \subset \Lambda \) with the following property: for any point \(y \in C,\) there exists a path \({{\psi }_{y}}:I \to \Delta \cup C\) such that \({{\psi }_{y}}(1) = y\) and \({{\psi }_{y}}(t) \in \Delta \) for any \(t \in [0,1)\).
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ACKNOWLEDGMENTS
This work was supported by the Russian Science Foundation, project no. 17-11-01041.
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Translated by I. Ruzanova
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Grines, V.Z., Kurenkov, E.D. Classification of One-Dimensional Attractors of Diffeomorphisms of Surfaces by Means of Pseudo-Anosov Homeomorphisms. Dokl. Math. 99, 137–139 (2019). https://doi.org/10.1134/S1064562419020066
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DOI: https://doi.org/10.1134/S1064562419020066