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January  2000, 6(1): 61-88. doi: 10.3934/dcds.2000.6.61

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, United States
2. 

Department of Mathematics, Penn State University, University Park, State College, PA 16802, United States

Received  October 1999 Published  December 1999

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.
Keywords: Conjugation-­approximation method, partition rearrangements, almost isometries, weakly mixing dioeeomorphisms, measure­preserving exponential map..
Mathematics Subject Classification: 58F11 (primary), 28D05, 57R50, 53C99 (secondary.
Citation: Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
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