Benardete, D. ; Dani, S. G.
(1999)
*Topological equivalence and rigidity of flows on certain solvmanifolds*
Ergodic Theory & Dynamical Systems, 19
(3).
pp. 559-569.
ISSN 0143-3857

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Official URL: http://journals.cambridge.org/action/displayAbstra...

Related URL: http://dx.doi.org/10.1017/S0143385799130116

## Abstract

Given a Lie group $G$ and a lattice $\Gamma$ in $G$, a one-parameter subgroup $\phi$ of $G$ is said to be rigid if for any other one-parameter subgroup $\psi$, the flows induced by $\phi$ and $\psi$ on $\Gamma\backslash G$ (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if $G$ is a simply connected solvable Lie group such that all the eigenvalues of $\mathrm{Ad} (g) $, $g\in G$, are real, then all one-parameter subgroups of $G$ are rigid for any lattice in $G$. Here we consider a complementary case, in which the eigenvalues of $\mathrm{Ad} (g)$, $g\in G$, form the unit circle of complex numbers. Let $G$ be the semidirect product $N \rtimes M$, where $M$ and $N$ are finite-dimensional real vector spaces and where the action of $M$ on the normal subgroup $N$ is such that the center of $G$ is a lattice in $M$. We prove that there is a generic class of abelian lattices $\Gamma$ in $G$ such that any semisimple one-parameter subgroup $\phi$ (namely $\phi$ such that $\mathrm{Ad} (\phi_t)$ is diagonalizable over the complex numbers for all $t$) is rigid for $\Gamma$ (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple $\phi$ are not rigid (see Corollary 4.3); further, there are non-rigid semisimple $\phi$ for which the induced flow is ergodic.

Item Type: | Article |
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Source: | Copyright of this article belongs to Cambridge University Press. |

ID Code: | 8261 |

Deposited On: | 26 Oct 2010 11:58 |

Last Modified: | 30 May 2011 06:02 |

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