Topological objects in invariant sets of dynamical systems

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 self-similar 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 2-sphere. A curve contained in the 2-torus 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 2-torus. 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 one-dimensional separatrix of Morse-Smale diffeomorphism on the three-dimensional 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. 17-11-01041.
References
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[2] Bonatti Ch., Grines V. Knots as topological invariant for gradient-like 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.