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On the arc connecting a rough diffeomorphism on a 3-torus with a expanding attractor and Anosov hyperbolic automorphism

TitleOn the arc connecting a rough diffeomorphism on a 3-torus with a expanding attractor and Anosov hyperbolic automorphism
AuthorsV. Z. Grines1, E. V. Kruglov2, O. V. Pochinka1
1National Research University Higher School of Economics
2National Research Lobachevsky State University of Nizhny Novgorod
AnnotationSmale'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.
KeywordsAnosov diffeomorphism, saddle-node bifurcation, flip bifurcation
CitationGrines 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: 04.12.2024.