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Stably ergodic approximation: Two examples. (English) Zbl 0970.37022
C. Pugh and M. Shub [J. Eur. Math. Soc. (JEMS) 2, No. 1, 1-52 (2000; Zbl 0964.37017)] recently conjectured that the stably ergodic diffeomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diffeomorphisms of a compact manifold. In the present paper they consider two examples: the standard map cross Anosov and the ergodic automorphisms of the 4-torus. In both cases they show that the maps in question may be approximated by stably ergodic diffeomorphisms which have the stable accessibility property. A sample result: For any \(C^\infty\) symplectic Anosov diffeomorphism \(f\), the map \(f\times g_\lambda\) can be \(C^\infty\) approximated arbitrarily well by a symplectic, stably ergodic diffeomorphism if \(\lambda\) is sufficiently close to zero. Here \(g_\lambda\) is the standard map of the 2-torus given by \(g_\lambda (z,w)=(z+w, w+\lambda\sin (2\pi (z+w)))\).

MSC:
37D30 Partially hyperbolic systems and dominated splittings
37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C20 Generic properties, structural stability of dynamical systems
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
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