  We propose an extension of ergodic theory which focuses on the identification
of ergodicity in terms of the uniqueness of the invariant measure. We first
explain the concept for the doubling maps, which can be analyzed using Fourier
methods. We then proceed to the Gauss-type maps of interest, of the form
$x\mapsto -\beta/x$ mod $2\mathbb Z$ on the symmetric interval $[-1,1]$, for
$0<\beta\le1$. We study an extended state space on the interval, formed as the
restriction to the interval $[-1,1]$ of functions of the form $f+\mathbf{H}g$,
where $f$ and $g$ are $L^1$-functions. We then look for invariant states for
the Gauss-type map. We find that the standard ergodicity results available for
$L^1$ extend with difficulty to the larger state space. The machinery developed
involves a dynamical decomposition of the odd part of the Hilbert kernel. We
apply the result to decide the issue when the nonnegative integer powers of two
given atomic singular inner functions is complete in $H^\infty$ with respect to
the weak-star topology.
