Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure

Roland Gunesch; Anatole Katok

doi:10.3934/dcds.2000.6.61

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2000, Volume 6, Issue 1: 61-88. Doi: 10.3934/dcds.2000.6.61

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Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure

Roland Gunesch^1, and
Anatole Katok^2,

1.
Department of Mathematics, Pennsylvania State University, University Park, PA 16802
2.
Department of Mathematics, Penn State University, University Park, State College, PA 16802

Received: September 30, 1999

Published: December 1999

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Abstract

We describe in detail a construction of weakly mixing $C^\infty$ diffeomorphisms preserving a smooth measure and a measurable Riemannian metric as well as ${\mathbb} Z^k$ actions with similar properties. We construct those as a perturbation of elements of a nontrivial non-transitive circle action. Our construction works on all compact manifolds admitting a nontrivial circle action.
It is shown in the appendix that a Riemannian metric preserved by a weakly mixing diffeomorphism can not be square integrable.

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Mathematics Subject Classification: 58F11 (primary), 28D05, 57R50, 53C99 (secondary).

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