American Institute of Mathematical Sciences

Advanced
April  2008, 2(2): 339-358. doi: 10.3934/jmd.2008.2.339

Ergodic properties of isoperimetric domains in spheres

Gerhard Knieper 1, and  Norbert Peyerimhoff 2,

1. 

Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany
2. 

Department of Math. Sciences, Durham University, Durham DH1 3LE, United Kingdom

Received  September 2007 Revised  December 2007 Published  January 2008

Let $\varphi$ be a function on the unit tangent bundle of a compact manifold of negative curvature. We show that averages of $\varphi$ over subdomains of increasing spheres converge to the horospherical mean if these domains satisfy an isoperimetric condition. We apply this result to spherical means with continuous density and, by using relations between the horospherical mean and the Patterson-Sullivan measure, we derive some kind of mixing properties.
Keywords: spherical means, mixing., geodesic flows.
Mathematics Subject Classification: Primary 37C40, Secondary 53C12, 37C1.
Citation: Gerhard Knieper, Norbert Peyerimhoff. Ergodic properties of isoperimetric domains in spheres. Journal of Modern Dynamics, 2008, 2 (2) : 339-358. doi: 10.3934/jmd.2008.2.339
[1]

Fei Liu, Xiaokai Liu, Fang Wang. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4791-4804. doi: 10.3934/dcds.2021057
[2]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[3]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841
[4]

Daniel Visscher. A new proof of Franks' lemma for geodesic flows. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4875-4895. doi: 10.3934/dcds.2014.34.4875
[5]

Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13
[6]

Chenchen Wu, Wei Lv, Yujie Wang, Dachuan Xu. Approximation algorithm for spherical $ k $-means problem with penalty. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021067
[7]

Vladislav Kruglov, Dmitry Malyshev, Olga Pochinka. Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4305-4327. doi: 10.3934/dcds.2018188
[8]

Jeffrey Boland. On rigidity properties of contact time changes of locally symmetric geodesic flows. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 645-650. doi: 10.3934/dcds.2000.6.645
[9]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477
[10]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
[11]

Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403
[12]

Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61
[13]

Ítalo Melo, Sergio Romaña. Contributions to the study of Anosov geodesic flows in non-compact manifolds. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5149-5171. doi: 10.3934/dcds.2020223
[14]

Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427
[15]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[16]

Krzysztof Frączek, M. Lemańczyk, E. Lesigne. Mild mixing property for special flows under piecewise constant functions. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 691-710. doi: 10.3934/dcds.2007.19.691
[17]

Adam Kanigowski, Davide Ravotti. Polynomial 3-mixing for smooth time-changes of horocycle flows. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5347-5371. doi: 10.3934/dcds.2020230
[18]

Krzysztof Frączek, Mariusz Lemańczyk. A class of mixing special flows over two--dimensional rotations. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4823-4829. doi: 10.3934/dcds.2015.35.4823
[19]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[20]

François Gay-Balmaz, Cesare Tronci, Cornelia Vizman. Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups. Journal of Geometric Mechanics, 2013, 5 (1) : 39-84. doi: 10.3934/jgm.2013.5.39

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy