Computing Invariant Measures of Piecewise Convex Maps with Countably Many Branches

In this paper, we develop convergent numerical schemes to approximate absolutely continuous invariant measures of piecewise convex maps with countably many branches, based on a modified piecewise-linear Markov method and a novel piecewise quadratic B-spline method. The proposed schemes are effective for studying piecewise convex chaotic maps. Notably, while it ensures the same convergence order as the piecewise-linear method, the piecewise quadratic B-spline method enhances the smoothness of the approximate density. Numerical examples demonstrate the efficacy of these approaches.