On bounded cocycles of isometries over minimal dynamics
Daniel Coronel - Departamento deMatemática, UNAB, República 220, 2 piso, Santiago, 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.
Received: June 2012; Revised: January 2013; Available Online: May 2013. |