  We consider two-sided Jacobi matrices whose coefficients are obtained by
continuous sampling along the orbits of a homeomorphim of a compact metric
space. Given an ergodic probability measure, we study the topological structure
of the associated almost sure spectrum. We establish a gap labelling theorem in
the spirit of Johnson and Schwartzman. That is, we show that the constant value
the integrated density of states takes in a gap of the spectrum must belong to
the countable Schwartzman group of the base dynamics. This result is a natural
companion to a recent result of Alkorn and Zhang, which established a
Johnson-type theorem for the families of Jacobi matrices in question.
