Two bifurcation sets of expansive Lorenz maps with a hole at the critical point

Let $f$ be an expansive Lorenz map on $[0,1]$ and $c$ be the critical point. The survivor set we are discussing here is denoted as $S^+_{f}(a,b):=\{x\in[0,1]:f(b)\leq f^{n}(x) \leq f(a)\ \forall n\geq0\}$, where the hole $(a,b)\subseteq [0,1]$ satisfies $a\leq c \leq b$ and $a\neq b$. Let $a\in[0,c]$ be fixed, we mainly focus on the following two bifurcation sets: $$E_{f}(a):=\{b\in[c,1]:S^{+}_{f}(a,\epsilon)\neq S^{+}_{f}(a,b) \ \forall \ \epsilon>b\}, \ \ {\rm and}$$ $$B_{f}(a):=\{b\in[c,1]:h_{top}(S^+_{f}(a,\epsilon))\neq h_{top}(S^+_{f}(a,b)) \ \forall \ \epsilon>b\}.$$ By combinatorial renormalization tools, we give a complete characterization of the maximal plateau $P(b)$ such that for all $\epsilon\in P(b)$, $h_{top}(S^+_{f}(a,\epsilon))=h_{top}(S^+_{f}(a,b))$. Moreover, we obtain a sufficient and necessary condition for $E_{f}(a)=B_{f}(a)$, which partially extends the results in \cite{allaart2023} and \cite{baker2020}.