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On bounded cocycles of isometries over minimal dynamics

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  • 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.
    Mathematics Subject Classification: Primary: 37A99; Secondary: 54H15.


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