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Due to a result by Glasner and Downarowicz, it is known that a minimal systemis mean equicontinuous if and only if it is an isomorphic extension of itsmaximal equicontinuous factor. The majority of known examples of this type arealmost automorphic, that is, the factor map to the maximal equicontinuousfactor is almost one-to-one. The only cases of isomorphic extensions which arenot almost automorphic are again due to Glasner and Downarowicz, who in thesame article provide a construction of such systems in a rather generaltopological setting.  Here, we use the Anosov-Katok method in order to provide an alternative routeto such examples and to show that these may be realised as smooth skew productdiffeomorphisms of the two-torus with an irrational rotation on the base.Moreover - and more importantly - a modification of the construction allows toensure that lifts of these diffeomorphism to finite covering spaces providenovel examples of finite-to-one topomorphic extensions of irrational rotations.These are still strictly ergodic and share the same dynamical eigenvalues asthe original system, but show an additional singular continuous component ofthe dynamical spectrum.

Construction of smooth isomorphic and finite-to-one extensions of irrational rotations which are not almost automorphic