Open dynamical systems in topologically expansive decreasing Lorenz maps

We develop a parity-sensitive admissibility theory for topologically expansive decreasing Lorenz maps. By embedding them into the standard circle map $m_{-2}$, we realize every decreasing-admissible pair. We then study the negative doubling map $T_{-2}$ with a hole $(a,b)$, where $0<a\le 1\mathrm{/}2\le b<1$. Using the alternating lexicographic order, we classify plateaux of the symbolic survivor spaces. Under weak admissibility and strict shifted ordering, the boundary subshifts are realized as kneading spaces of decreasing Lorenz maps. We also establish entropy correspondences across countable itinerary discrepancies in completed survivor systems. For fixed $a$, we prove that $b\mapsto {\dim }_{\mathrm{H}}{S}_{-2}(a,b)$ is a devil’s staircase, possibly constant. Under the pullback-null condition~(P), we obtain the corresponding survivor-entropy result for expansive decreasing Lorenz maps. We verify this condition for the piecewise-linear family in Example 5.1.