Diffeomorphisms of 3-manifolds with 1-dimensional basic sets exteriorly situated on 2-tori

V. Z. Grines ; O. V. Pochinka ; A. A. Shilovskaya

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 18 (2016) no. 1, pp. 17-26

Résumé

In this paper we consider the class $G$ of A-diffeomorphisms $f$, defined on a closed 3-manifold $M^3$. The nonwandering set is located on finite number of pairwise disjoint $f$-invariant 2-tori embedded in $M^3$. Each torus $T$ is a union of $W^u_{B_T}\cup W^u_{\Sigma_T}$ or $W^s_{B_T}\cup W^s_{\Sigma_T}$, where $B_T$ is 1-dimensional basic set exteriorly situated on $T$ and $\Sigma_T$ is finite number of periodic points with the same Morse number. We found out that an ambient manifold which allows such diffeomorphisms is homeomorphic to a quotient space $M_{\widehat J}=\mathbb T^2\times[0,1]/_\sim$, where $(z,1)\sim(\widehat J(z),0)$ for some algebraic torus automorphism $\widehat J$, defined by matrix $J\in GL(2,\mathbb Z)$ which is either hyperbolic or $J=\pm Id$. We showed that each diffeomorphism $f\in G$ is semiconjugate to a local direct product of an Anosov diffeomorphism and a rough circle transformation. We proved that structurally stable diffeomorphism $f\in G$ is topologically conjugate to a local direct product of a generalized DA-diffeomorphism and a rough circle transformation. For these diffeomorphisms we found the complete system of topological invariants; we also constructed a standard representative in each class of topological conjugation.

Détail

Keywords: А-diffeomorphism, DA-diffeomorphism, topological invariant, topological conjugation.