Abstract:
In this paper we consider an $\Omega$-stable $3$-diffeomorphism whose chain-recurrent set consists of isolated periodic points and hyperbolic $2$-dimensional nontrivial attractors. Nontrivial attractors in this case can only be expanding, orientable or not. The most known example from the class under consideration is the DA-diffeomorphism obtained from the algebraic Anosov diffeomorphism, given on a $3$-torus, by Smale’s surgery. Each such attractor has bunches of degree $1$ and $2$. We estimate the minimum number of isolated periodic points using information about the structure of attractors. Also, we investigate the topological structure of ambient manifolds for diffeomorphisms with $k$ bunches and $k$ isolated periodic points.
Keywords:
hyperbolicity, expanding attractor, $\Omega$-stability, nonwandering set, regular system
This article is an output of a research project implemented as part of the Basic Research Program
at the National Research University Higher School of Economics (HSE University).
\Bibitem{Bar24}
\by Marina K. Barinova
\paper On Isolated Periodic Points of Diffeomorphisms with Expanding Attractors of Codimension $1$
\jour Regul. Chaotic Dyn.
\yr 2024
\vol 29
\issue 5
\pages 794--802
\mathnet{http://mi.mathnet.ru/rcd1282}
\crossref{https://doi.org/10.1134/S1560354724050022}
Linking options:
https://www.mathnet.ru/eng/rcd1282
https://www.mathnet.ru/eng/rcd/v29/i5/p794
This publication is cited in the following 3 articles:
Marina K. Barinova, Evgenii M. Osenkov, Olga V. Pochinka, “On Morse – Smale 3-Diffeomorphisms with a Given Tuple of Sink Points Periods”, Regul. Chaotic Dyn., 30:2 (2025), 226–253
M. K. Barinova, O. A. Kolchurina, E. I. Yakovlev, “On 3-diffeomorphisms with generalized Plykin attractor”, Mat. Sb., 215:9 (2024), 3–29
M. K. Barinova, O. A. Kolchurina, E. I. Yakovlev, “On 3-diffeomorphisms with generalized Plykin attractors”, Sb. Math., 215:9 (2024), 1135–1158
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