Let (B,T) be a fibred system, where B is a compact and convex subset of \mathbb{R}^{d} and T:B\mapsto B is a map. Under adequate norm the space \mathcal{L} (\mathcal{H}) (respectively \mathcal{C} (\mathcal{H})) of Lipschitz continuous (respectively continuous) functions on every cell H of a certain finite partition \mathcal{H} of B is a Banach space. We delimit by Conditions \rm (A),\dotsc,(E),(F)' a class of fibred systems, including multidimensional continued fractions. We first prove that the Perron–Frobenius operator A associated with T under the Lebesgue measure \lambda considered on <formula form="inline" disc="math" id="eq014"><formtex notation="amstex">\mathcal{L} (\mathcal{H}) is weak-mixing. Using the ergodic theorem of Ionescu–Tulcea and Marinescu we then prove that there exist positive constants q >1 and M for which

\vert\mspace{-2mu}\vert\mspace{-2mu}\vert A^{n}\varphi -h\int_{B}\varphi \,d\lambda\vert\mspace{-2mu}\vert\mspace{-2mu}\vert \leq q^{n}M\vert\mspace{-2mu}\vert\mspace{-2mu}\vert\varphi \vert\mspace{-2mu}\vert\mspace{-2mu}\vert, \quad n\in \mathbb{N}, \varphi \in \mathcal{L} (\mathcal{H}),

where \vert\mspace{-2mu}\vert\mspace{-2mu}\vert\cdot\vert\mspace{-2mu}\vert\mspace{-2mu}\vert is the norm in \mathcal{L} (\mathcal{H}) and h is the density of the (unique) T-invariant probability \mu on the Borel \sigma-algebra \Sigma in B. The analogous result for the Perron–Frobenius operator U of T under \mu is also proved.