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 & Continuous Dynamical Systems, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61
[1]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817

[2]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[3]

Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067

[4]

Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027

[5]

Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013

[6]

Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079

[7]

François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275

[8]

Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353

[9]

S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343

[10]

Sorin Micu, Ademir F. Pazoto. Almost periodic solutions for a weakly dissipated hybrid system. Mathematical Control & Related Fields, 2014, 4 (1) : 101-113. doi: 10.3934/mcrf.2014.4.101

[11]

Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012

[12]

Wacław Marzantowicz, Justyna Signerska. Firing map of an almost periodic input function. Conference Publications, 2011, 2011 (Special) : 1032-1041. doi: 10.3934/proc.2011.2011.1032

[13]

Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011

[14]

Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017

[15]

Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645

[16]

Xuhui Peng, Jianhua Huang, Yan Zheng. Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4479-4506. doi: 10.3934/cpaa.2020204

[17]

Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307

[18]

James Benn. Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group. Journal of Geometric Mechanics, 2016, 8 (1) : 1-12. doi: 10.3934/jgm.2016.8.1

[19]

Sahani Pathiraja, Wilhelm Stannat. Analysis of the feedback particle filter with diffusion map based approximation of the gain. Foundations of Data Science, 2021, 3 (3) : 615-645. doi: 10.3934/fods.2021023

[20]

Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]