Journal of Modern Dynamics (JMD)

On bounded cocycles of isometries over minimal dynamics

Pages: 45 - 74, Issue 1, March 2013      doi:10.3934/jmd.2013.7.45

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Daniel Coronel - Departamento deMatemática, UNAB, República 220, 2 piso, Santiago, Chile (email)
Andrés Navas - Departamento de Matemática y C.C., USACH, Alameda 3363, Estación Central, Santiago, Chile (email)
Mario Ponce - Facultad deMatemáticas, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile (email)

Abstract: We show the following geometric generalization of a classical theorem of W. H. Gottschalk and G. A. Hedlund: a skew action induced by a cocycle of (affine) isometries of a Hilbert space over a minimal dynamical system has a continuous invariant section if and only if the cocycle is bounded. Equivalently, the associated twisted cohomological equation has a continuous solution if and only if the cocycle is bounded. We interpret this as a version of the Bruhat-Tits Center Lemma in the space of continuous functions. Our result also holds when the fiber is a proper CAT(0) space. One of the applications concerns matrix cocycles. Using the action of $\mathrm{GL} (n,\mathbb{R})$ on the (nonpositively curved) space of positively definite matrices, we show that every bounded linear cocycle over a minimal dynamical system is cohomologous to a cocycle taking values in the orthogonal group.

Keywords:  Cohomological equation, cocycle, CAT(0) space.
Mathematics Subject Classification:  Primary: 37A99; Secondary: 54H15.

Received: June 2012;      Revised: January 2013;      Available Online: May 2013.