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 CMFD, 2015, Volume 57, Pages 5–30 (Mi cmfd270)

Rough diffeomorphisms with basic sets of codimension one

V. Z. Grinesab, Ye. V. Zhuzhomab, O. V. Pochinkab

a Lobachevski State University of Nizhni Novgorod
b National Research University "Higher School of Economics", Nizhny Novgorod Branch

Abstract: The review is devoted to the exposition of results (including those of the authors of the review) obtained from the 2000s until the present, on topological classification of structurally stable cascades defined on a smooth closed manifold $M^n$ ($n\ge3$) assuming that their nonwandering sets either contain an orientable expanding (contracting) attractor (repeller) of codimension one or completely consist of basic sets of codimension one. The results presented here are a natural continuation of the topological classification of Anosov diffeomorphisms of codimension one. The review also reflects progress related to construction of the global Lyapunov function and the energy function for dynamical systems on manifolds (in particular, a construction of the energy function for structurally stable $3$-cascades with a nonwandering set containing a two-dimensional expanding attractor is described).

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English version:
Journal of Mathematical Sciences, 2017, 225:2, 195–219

UDC: 517.938

Citation: V. Z. Grines, Ye. V. Zhuzhoma, O. V. Pochinka, “Rough diffeomorphisms with basic sets of codimension one”, Proceedings of the Crimean autumn mathematical school-symposium, CMFD, 57, PFUR, M., 2015, 5–30; Journal of Mathematical Sciences, 225:2 (2017), 195–219

Citation in format AMSBIB
\Bibitem{GriZhuPoc15} \by V.~Z.~Grines, Ye.~V.~Zhuzhoma, O.~V.~Pochinka \paper Rough diffeomorphisms with basic sets of codimension one \inbook Proceedings of the Crimean autumn mathematical school-symposium \serial CMFD \yr 2015 \vol 57 \pages 5--30 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd270} \transl \jour Journal of Mathematical Sciences \yr 2017 \vol 225 \issue 2 \pages 195--219 \crossref{https://doi.org/10.1007/s10958-017-3468-3} 

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This publication is cited in the following articles:
1. V. Z. Grines, O. V. Pochinka, “Postroenie energeticheskikh funktsii dlya $\Omega$-ustoichivykh diffeomorfizmov na $2$- i $3$-mnogoobraziyakh”, Trudy Krymskoi osennei matematicheskoi shkoly-simpoziuma, SMFN, 63, no. 2, Rossiiskii universitet druzhby narodov, M., 2017, 191–222
2. V. Z. Grines, E. V. Kruglov, O. V. Pochinka, “Scenario of a Simple Transition from a Structurally Stable 3-Diffeomorphism with a Two-Dimensional Expanding Attractor to a DA Diffeomorphism”, Proc. Steklov Inst. Math., 308 (2020), 141–154
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