Uniqueness of large invariant measures  for  $\mathbb{Z}^k$ actions with Cartan homotopy data

Anatole Katok; Federico Rodriguez Hertz

doi:10.3934/jmd.2007.1.287

Article Contents

2007, Volume 1, Issue 2: 287-300. Doi: 10.3934/jmd.2007.1.287

This issue Previous Article Prequantum chaos: Resonances of the prequantum cat map Next Article Smooth Anosov flows: Correlation spectra and stability

Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data

Anatole Katok^1, and
Federico Rodriguez Hertz^2,

1.
Department of Mathematics, The Pennsylvania State University, University Park, PA 16802
2.
IMERL-Facultad de Ingeniería, Universidad de la República, ulio Herrera y Reissig 565, CC 30, 11300 Montevideo

Received: May 2006

Revised: December 2006

Published: January 2007

Abstract / Introduction Related Papers Cited by

Abstract

Every $C^2$ action $\a$ of $\mathbb{Z}k$, $k\ge 2$, on the $(k+1)$-dimensional torus whose elements are homotopic to the corresponding elements of an action $\ao$ by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between $\a$ and $\a_0$. This measure is absolutely continuous and the semiconjugacy provides a measure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for $\a$ for which the eigenvalues for $\a$ and $\a_0$ coincide. We describe some nontrivial examples of actions of this kind.

Keywords:

Mathematics Subject Classification: Primary: 37C40, 37D25, 37C85.

Citation:

Article Metrics

HTML views() PDF downloads(62) Cited by(0)

Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content