On bounded cocycles of isometries over a minimal dynamics

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 dynamics 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 appplications 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 dynamics is cohomologous to a cocycle taking values in the orthogonal group.