An open dense set of stably ergodic diffeomorphisms in a neighborhood of a non-ergodic one

By V. Nitica, Andrei Török. 2001

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Accessibility (2018, unknown): “Arbitrarily near any product of an Anosov diffeomorphism and a circle rotation, there exists a C 1 -open set of accessible yet non-transitive partially hyperbolic diffeomorphisms [NtaT01], which therefore have wandering points due to Brin’s result [Bri75].”
Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle (2006, background): “In [ 24 ], Ni¸ticu nd A. Torok proved stable accessibility is C r dense for one-dimensional center bundle, under certain hypotheses (for instance, dynamical coherence and compact center leaves).”
Maximizing measures for partially hyperbolic systems with compact center leaves (2010, background): “But accessibility is an open and dense property in this context (see [25, 6]).”