2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127876We 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.25 pagesDynamical Systems37C30, 37D20text