The destruction of the Smale-Williams solenoids
S. V. Gonchenko1, E.V. Zhuzhoma2, N.V. Isaenkova3
| Annotation | In the paper, one represents the family of diffeomorphisms $f_{\nu}: S^3\to S^3$, $-1\leq\nu\leq 1$, depending smoothly on the parameter $\nu$ such that 1) given any $-1\leq\nu <0$, the non-wandering set of $f_{\nu}$ consists of one-dimensional expanding attractor and one-dimensional contracting repeller that are Smale-Williams solenoid; 2) the diffeomorphism $f_0$ has a non-wandering set consisting of the two zero-dimensional transitive invariant sets $\Lambda_1$ and $\Lambda_2$ such that each is homeomorphic to the product of Cantor sets, and the restriction $f_0|_{\Lambda_1\cup\Lambda_2}$ is a partially hyperbolic diffeomorphism; 3) given any $0<\nu\leq 1$, the non-wandering set of $f_{\nu}$ consists of two hyperbolic zero-dimensional transitive invariant sets each is homeomorphic to the product of Cantor sets. |
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| Keywords | Attractor, repeller, solenoid Smale-Williams |
1Head of Department of Differential Equations, Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod
2Professor of mathematical analysis, theory and methodology of training, Nizhny Novgorod State Pedagogical University, Nizhny Novgorod; zhuzhoma@mail.ru.
3Senior Lecturer, Department of Mathematical Analysis, Nizhny Novgorod State Pedagogical University, Nizhny Novgorod; zhuzhoma@mail.ru.
Citation: S. V. Gonchenko, E.V. Zhuzhoma, N.V. Isaenkova , "[The destruction of the Smale-Williams solenoids]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,15:1 (2013) 65–70 (In Russian)
