# American Institute of Mathematical Sciences

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

 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.
Citation: Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
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