Stable Arcs Connecting Polar Cascades on a Torus

    Received 28 February 2021

    2021, Vol. 17, no. 1, pp.  23-37

    Author(s): Pochinka O. V., Nozdrinova E. V.

    The problem of the existence of an arc with at most countable (finite) number of bifurcations connecting structurally stable systems (Morse – Smale systems) on manifolds was included in the list of fifty Palis – Pugh problems at number 33.
    In 1976 S. Newhouse, J.Palis, F.Takens introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its quality properties with small changes. In the same year, S.Newhouse and M.Peixoto proved the existence of a simple arc (containing only elementary bifurcations) between any two Morse – Smale flows. From the result of the work of J. Fliteas it follows that the simple arc constructed by Newhouse and Peixoto can always be replaced by a stable one. For Morse – Smale diffeomorphisms defined on manifolds of any dimension, there are examples of systems that cannot be connected by a stable arc. In this connection, the question naturally arises of finding an invariant that uniquely determines the equivalence class of a Morse – Smale diffeomorphism with respect to the relation of connection by a stable arc (a component of a stable isotopic connection).
    In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.
    Keywords: stable arc, saddle-node, gradient-like diffeomorphism, two-dimensional torus
    Citation: Pochinka O. V., Nozdrinova E. V., Stable Arcs Connecting Polar Cascades on a Torus, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 1, pp.  23-37

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