  We consider suspension semi-flows of angle-multiplying maps on the circle.
Under a $C^r$generic condition on the ceiling function, we show that there
exists an anisotropic Sobolev space contained in the $L^2$ space such that the
Perron-Frobenius operator for the time-$t$-map acts on it and that the
essential spectral radius of that action is bounded by the square root of the
inverse of the minimum expansion rate. This leads to a precise description on
decay of correlations and extends the result of M. Pollicott.
