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We study the dynamics of Euclidean isometric extensions of minimal homeomorphisms of compact metric spaces. Under a general hypothesis of homogeneity for the base space, we show that these systems are never minimal, thus extending a classical result of Besicovitch concerning cylindrical cascades. Moreover, using Anosov–Katok‐type methods, we construct a topologically transitive isometric extension over an irrational rotation with a two‐dimensional fiber.

On the dynamics of nonreducible cylindrical vortices