On the Topological Conjugacy of Skew Products with a Chaotic Base

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.