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 Trudy Mat. Inst. Steklova, 2004, Volume 244, Pages 143–215 (Mi tm446)

Combinatorics of One-Dimensional Hyperbolic Attractors of Diffeomorphisms of Surfaces

A. Yu. Zhirov

Abstract: An algorithmic solution is given to the two following problems. Let $\Lambda _f$ and $\Lambda _g$ be one-dimensional hyperbolic attractors of diffeomorphisms $f\colon M\to M$ and $g\colon N\to N$, where $M$ and $N$ are closed surfaces, either orientable or not. Does there exist a homeomorphism $h\colon U(\Lambda _f)\to V(\Lambda _g)$ of certain neighborhoods of attractors such that $f\circ h=h\circ g$ (the topological conjugacy problem). Given $h>0$, find a representative of each class of topological conjugacy of attractors with a given structure of accessible boundary (boundary type) for which topological entropy is no greater than $h$ (the problem of enumeration of attractors). The solution of these problems is based on the combinatorial method, developed by the author, for describing hyperbolic attractors of surface diffeomorphisms.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 244, 132–200

Bibliographic databases:
UDC: 517.938.5

Citation: A. Yu. Zhirov, “Combinatorics of One-Dimensional Hyperbolic Attractors of Diffeomorphisms of Surfaces”, Dynamical systems and related problems of geometry, Collected papers. Dedicated to the memory of academician Andrei Andreevich Bolibrukh, Trudy Mat. Inst. Steklova, 244, Nauka, MAIK «Nauka/Inteperiodika», M., 2004, 143–215; Proc. Steklov Inst. Math., 244 (2004), 132–200

Citation in format AMSBIB
\Bibitem{Zhi04} \by A.~Yu.~Zhirov \paper Combinatorics of One-Dimensional Hyperbolic Attractors of Diffeomorphisms of Surfaces \inbook Dynamical systems and related problems of geometry \bookinfo Collected papers. Dedicated to the memory of academician Andrei Andreevich Bolibrukh \serial Trudy Mat. Inst. Steklova \yr 2004 \vol 244 \pages 143--215 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm446} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2075116} \zmath{https://zbmath.org/?q=an:1079.37036} \transl \jour Proc. Steklov Inst. Math. \yr 2004 \vol 244 \pages 132--200 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Grines V.Z., Zhuzhoma E.V., “Expanding attractors”, Regul. Chaotic Dyn., 11:2 (2006), 225–246
2. A. Yu. Zhirov, “How Many Different Cascades on a Surface Can Have Coinciding Hyperbolic Attractors?”, Math. Notes, 94:1 (2013), 96–106
3. A. G. Fedotov, “On the Realization of the Generalized Solenoid as a Hyperbolic Attractor of Sphere Diffeomorphisms”, Math. Notes, 94:5 (2013), 681–691
4. A. G. Fedotov, “On the Solenoidal Representation of the Hyperbolic Attractor of a Diffeomorphism of the Sphere”, Math. Notes, 101:1 (2017), 181–183
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