Vladislav Medvedev
Publications:
Medvedev V. S., Zhuzhoma E. V.
On a Classification of Chaotic Laminations which are Nontrivial Basic Sets of Axiom A Flows
2023, Vol. 19, no. 2, pp. 227237
Abstract
We prove that, given any $n\geqslant 3$ and $2\leqslant q\leqslant n1$, there is a closed $n$manifold $M^n$ admitting a chaotic lamination of codimension $q$ whose support has the topological dimension ${nq+1}$. For $n=3$ and $q=2$, such chaotic laminations can be represented as nontrivial 2dimensional basic sets of axiom A flows on 3manifolds. We show that there are two types of compactification (called casings) for a basin of a nonmixing 2dimensional basic set by a finite family of isolated periodic trajectories. It is proved that an axiom A flow on every casing has repellerattractor 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 onedimensional basic sets obtained previously by V. Grines, R. Plykin and Yu. Zhirov [16, 28, 31].
