

Dynamics in Siberia  2019
February 28, 2019 12:50–13:40, Novosibirsk, Sobolev Institute of Mathematics of Russian Academy of Sciences, Conference Hall





Plenary talks


Topological objects in invariant sets of dynamical systems
O. V. Pochinka^{} 
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Abstract:
Various topological constructions naturally emerge in the modern theory of dynamical systems. For instance, the Cantor set, discovered as an example of a set with cardinality of the continuum and zero Lebesgue measure, clarified the structure of expanding attractors and contracting repellers. Fractals, being selfsimilar objects with fractional dimension, are naturally found in complex dynamics. For example, the basin boundary of an attracting point can be the Julia set. The lakes of Wada, showing the phenomenon of a curve dividing the plane into more than two domains, were used in the construction of the Plykin attractor on the 2sphere. A curve contained in the 2torus and having an irrational winding number, being an injectively immersed subset but not a topological submanifold, was realized as an invariant manifold of a fixed point of the Anosov diffeomorphism of the 2torus. The Artin–Fox arc [1] and the mildly wild frame of Debruner–Fox arcs [3], symbolizing a wild set of hand arcs in $\mathbb R^3$, are realized by a frame of onedimensional separatrix of MorseSmale diffeomorphism on the threedimensional sphere [4], [2], [5]. These parallels can be continued for quite a long time, and this report is devoted to the construction of dynamic elements based on known topological objects.
Thanks. This work was supported by the grant of the Russian Science Foundation, grant no. 171101041.
References
[1] Artin E., Fox R. Some wild cells and spheres in threedimensional space // Ann. Math. 1948. V. 49. 979–990.
[2] Bonatti Ch., Grines V. Knots as topological invariant for gradientlike diffeomorphisms of the sphere $\mathbb S^3$ // Journal of Dynamical and Control Systems (Plenum Press, New York and London). 2000. V. 6. N. 4. 579–602.
[3] Debrunner H., Fox R. A mildly wild imbedding of an $n$frame. Duke Math. J. 27 (1960), no. 3, 425–429.
[4] D. Pixton. Wild unstable manifolds // Topology. 1977. V. 16. N. 2. 167–172.
[5] O.Pochinka. Diffeomorphisms with mildly wild frame of separatrices. Universitatis Iagelonicae Acta Mathematica, Fasciculus XLVII, 2009, 149–154.
Language: English

