Marina Barinova

The present paper gives a partial answer to Smale's question which diagrams can correspond to $(A,B)$-diffeomorphisms. Model diffeomorphisms of the two-dimensional torus derived by ``Smale surgery'' are considered, and necessary and sufficient conditions for their topological conjugacy are found. Also, a class $G$ of $(A,B)$-diffeomorphisms on surfaces which are the connected sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$-conjugated are constructed. Moreover, a subset $G_{*}^{} \subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$-conjugacy is singled out.

This paper investigates the topological conjugacy of skew products defined on the total space of locally trivial fiber bundles. We prove that under certain restrictions on the chaotic dynamics of the homeomorphism on the base, namely, the inability to choose a closed connected subset of the supporting manifold which is also an invariant set of the base homeomorphism (indecomposable chaos), any homeomorphism conjugating the skew products must also be the skew product. Thus, the topological conjugacy of homeomorphisms on the base spaces is a necessary condition for the conjugacy of such skew products. As a consequence, we establish that the direct products of indecomposably chaotic homeomorphisms with homeomorphisms having finite $\omega$-limit sets are topologically conjugate if and only if they are conjugate componentwise.