Let f be an expansive Lorenz map and c be the critical point. The survivor set is denoted as \(S_{f}(H):=\{x\in [0,1]: f^{n}(x)\notin H, \forall n\ge 0\}\) , where H is an open subinterval. Here we study the hole \(H=(a,b)\) with \(a\le c \le b\) and \(a\ne b \) . We observe that the case \(a=c\) is equivalent to the hole at 0, and the case \(b=c\) is equivalent to the hole at 1. Given any expansive Lorenz map f with a hole \(H=(a,b)\) and \(S_{f}(H)\nsubseteqq \{0,1\}\) , we prove that there exists a Lorenz map g such that \(\tilde{S}_{f}(H)\setminus \Omega (g)\) is countable, where \(\Omega (g)\) is the Lorenz-shift of g and \(\tilde{S}_{f}(H)\) is the symbolic representation of \(S_{f}(H)\) . Moreover, let a be fixed, we also give a complete characterization of the maximal plateau I(b) such that for all \(\epsilon \in I(b)\) , \(S^+_{f}(a,\epsilon )=S^+_{f}(a,b)\) , and I(b) may degenerate to a single point b. As an application, when f has an ergodic acim and a is fixed, we obtain that the topological entropy function \(\lambda _{f}(a):b\mapsto h_{top}(f|S_{f}(a,b))\) is a devil staircase. At the special case that f being an intermediate \(\beta \) -transformation, using the Ledrappier-Young formula, the Hausdorff dimension function \(\eta _{f}(a):b\mapsto \dim _{\mathcal {H}}(S_{f}(a,b))\) is naturally a devil staircase when fixing a. All the results can be naturally extended to the case that b is fixed. As a result, we extend the devil staircases in (Kalle et al. in Ergod Th Dyn Syst 40:2482–2514, 2020; Langeveld and Samuel in Acta Math Hungar 170:269–301, 2023; Urbanski in Ergod Th Dyn Syst 6:295–309, 1986) to expansive Lorenz maps with a hole at critical point.
