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

Previous Article
Infinite dimensional complex dynamics: Quasiconjugacies, localization and quantum chaos
DCDS Home
This Issue
Next Article
Expansiveness, specification, and equilibrium states for random bundle transformations

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

Abstract
Related Papers

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 - A, 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 - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[2]	Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067
[3]	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
[4]	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
[5]	Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079
[6]	François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275
[7]	Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353
[8]	S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343
[9]	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
[10]	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
[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]	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
[13]	Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017
[14]	Denis Volk. Almost every interval translation map of three intervals is finite type. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2307-2314. doi: 10.3934/dcds.2014.34.2307
[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]	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
[17]	Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.
[18]	Andrea Tosin, Paolo Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Networks & Heterogeneous Media, 2011, 6 (3) : 561-596. doi: 10.3934/nhm.2011.6.561
[19]	Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure & Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721
[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

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences