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

V. S. Medvedev ; E. V. Zhuzhoma

Russian journal of nonlinear dynamics, Tome 19 (2023) no. 2, pp. 227-237

Résumé

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].

Détail

Keywords: chaotic lamination, basic set, axiom A flow.