Homological reduction and entropy of discontinuous interval maps

We develop a homological approach to the topological entropy of discontinuous interval maps with arbitrary branch orientations. Relations among the one-sided images of discontinuities reduce the kneading determinant, while symmetries yield further simplifications. We determine the smallest possible dimension of this reduction for each orientation pattern and show that this minimum can coexist with maximal entropy. The general theory is applied to maps with increasing or decreasing branches, extending earlier results for bimodal maps and yielding exact formulas for entropy and periodic-orbit counts.