Statistical Stability and Linear Response for Suspension Measures under Base and Roof Perturbations

We study the dependence of canonical invariant probability measures of suspension semiflows on simultaneous perturbations of the invariant base density and the roof function. We compare two natural representations of the suspension measure: physical-height coordinates, in which the support depends on the roof function, and normalised-height coordinates, in which all measures are defined on a fixed cylinder. In physical-height coordinates, we obtain an exact fibrewise formula for the $L^1$ discrepancy between suspension densities and derive quantitative total-variation estimates. In normalised-height coordinates, we establish corresponding stability estimates through the product of the base density and the roof function. These results transfer quantitative statistical-stability estimates for the base densities to the associated suspension measures. For one-parameter families whose base densities are differentiable in $L^1$ and whose roof functions are differentiable in $L^\infty$, we derive an $L^1$ linear-response formula on the fixed cylinder and a bounded-Lipschitz first-variation formula in physical height. The latter contains a signed measure supported on the graph of the unperturbed roof. We further compute the exact first-order total-variation speed and prove that total-variation differentiability holds if and only if the first-order roof velocity vanishes almost everywhere with respect to the unperturbed fibre density. The results are measure-theoretic and do not require spectral or mixing assumptions on the suspension semiflow.