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On the arc connecting a rough diffeomorphism on a 3-torus with a expanding attractor and Anosov hyperbolic automorphism
| Title | On the arc connecting a rough diffeomorphism on a 3-torus with a expanding attractor and Anosov hyperbolic automorphism |
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| Authors | V. Z. Grines1, E. V. Kruglov2, O. V. Pochinka1 1National Research University Higher School of Economics 2National Research Lobachevsky State University of Nizhny Novgorod |
| Annotation | Smale's surgery on a three-dimensional torus makes it possible to obtain a so-called DA-diffeomorphism from an Anosov automorphism of codimension 1. Moreover, in the classical model, the DA-diffeomorphism has a single nontrivial basis set, which is a two-dimensional expanding attractor, and the remaining basis sets are trivial sources. The dynamics of an arbitrary structurally stable 3-diffeomorphism with such a nontrivial basis set is a generalization of the dynamics of classical DA-diffeomorphism: a generalized DA-diffeomorphism, like the classical one, exists only on a three-dimensional torus and has a single nontrivial basis set, while a saddle orbit can also be a trivial basis set of such a diffeomorphism, in addition to the source one. However, the arc of diffeomorphisms corresponding to Smale's surgical operation is not even mildly stable. By S. Newhouse, J. Palis and F. Thus, a hypothesis was proposed about the construction of a mildly stable arc between the Anosov diffeomorphism and the $DA$-diffeomorphism. In this paper, we discuss the construction of a mildly stable arc passing through simple saddle-node or flip bifurcations, connecting a structurally stable 3-diffeomorphism with a two-dimensional expanding attractor and an Anosov hyperbolic automorphism. |
| Keywords | Anosov diffeomorphism, saddle-node bifurcation, flip bifurcation |
| Citation | Grines V. Z., Kruglov E. V., Pochinka O. V. ''On the arc connecting a rough diffeomorphism on a 3-torus with a expanding attractor and Anosov hyperbolic automorphism'' [Electronic resource]. Proceedings of the XVI International scientific conference "Differential equations and their applications in mathematical modeling". (Saransk, July 17-20, 2023). Saransk: SVMO Publ, 2023. - pp. 48-53. Available at: https://conf.svmo.ru/files/2023/papers/paper06.pdf. - Date of access: 19.05.2024. |
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