On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows

    Received 17 January 2023; accepted 12 April 2023; published 27 April 2023

    2023, Vol. 19, no. 2, pp.  227-237

    Author(s): Medvedev V. S., Zhuzhoma E. V.

    We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n-1$, there is a closed $n$-manifold $M^n$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${n-q+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2-dimensional basic sets of axiom A flows on 3-manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2-dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repeller-attractor dynamics. For the first type of casing, the isolated periodic trajectories form a fibered link. The second type of casing is a locally trivial fiber bundle over a circle. In the latter case, we classify (up to neighborhood equivalence) such nonmixing basic sets on their casings with solvable fundamental groups. To be precise, we reduce the classification of basic sets to the classification (up to neighborhood conjugacy) of surface diffeomorphisms with one-dimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].
    Keywords: chaotic lamination, basic set, axiom A flow
    Citation: Medvedev V. S., Zhuzhoma E. V., On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows, Rus. J. Nonlin. Dyn., 2023, Vol. 19, no. 2, pp.  227-237

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