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 $S_f^{+}(a,b):=\{x\in [0,1]:f(b)\le f^n(x)\le f(a)\mathrm{\forall }n\ge 0\}$, where the hole $(a,b)\subseteq [0,1]$ satisfies $a\le c\le b$ and $a\mathrm{\ne }b$. Let $a\in [0,c]$ be fixed, we mainly focus on the following two bifurcation sets: $$E_f(a):=\{b\in [c,1]:S_f^{+}(a,\epsilon )\mathrm{\ne }{S}_f^{+}(a,b)\mathrm{\forall }\epsilon >b\},\mathrm{a}\mathrm{n}\mathrm{d}$$ $$B_f(a):=\{b\in [c,1]:h_{top}(S_f^{+}(a,\epsilon ))\mathrm{\ne }{h}_{top}(S_f^{+}(a,b))\mathrm{\forall }\epsilon >b\}\mathrm{.}$$ 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}.