Two Bifurcation Sets of Expansive Lorenz Maps with a Hole at the Discontinuity

Consider an expansive Lorenz map f defined on \(I=[0,1]\) $I=[0,1]$ and let c be its discontinuity. The survivor set under consideration is represented as \(\begin{aligned} S^+_{f}(a,b):=\{x\in I:f(b)\le f^{n}(x) \le f(a),\ \forall \ n\ge 0\}, \end{aligned}\) $$\begin{array}{c}\hfill S_f^{+}(a,b):=\{x\in I:f(b)\le f^n(x)\le f(a),\forall n\ge 0\},\end{array}$$ where \((a,b)\subseteq I\) $(a,b)\subseteq I$ satisfies \(a\le c \le b\) $a\le c\le b$ and \(a\ne b\) $a\ne b$ . We mainly study the following two bifurcation sets related to survivor set and its topological entropy \(h_{top}\) $h_{\mathit{top}}$ : \(\begin{aligned} E_{f}(a):= & \{b\ge c:S^{+}_{f}(a,\epsilon )\ne S^{+}_{f}(a,b), \ \forall \ \epsilon>b\}, \ \ {\textrm{and}}\\ B_{f}(a):= & \{b\ge c:h_{top}(f|_{S^+_{f}(a,\epsilon )})\ne h_{top}(f|_{S^+_{f}(a,b)}), \ \forall \ \epsilon >b\}. \end{aligned}\) $$\begin{array}{cc}\hfill E_f(a):=& \{b\ge c:S_f^{+}(a,ϵ)\ne S_f^{+}(a,b),\forall ϵ>b\},\text{and}\hfill \\ \hfill B_f(a):=& \{b\ge c:h_{\mathit{top}}(f{|}_{S_f^{+}(a,ϵ)})\ne h_{\mathit{top}}(f{|}_{S_f^{+}(a,b)}),\forall ϵ>b\}.\hfill \end{array}$$ By utilizing combinatorial renormalization techniques, we give the kneading sequences of the endpoints of the platform \(\begin{aligned} P(b):=\{b^\prime \ge c:h_{top}(f|_{S^{+}_{f}(a,b^\prime )})=h_{top}( f|_{S^{+}_{f}(a,b)})\}. \end{aligned}\) $$\begin{array}{c}\hfill P(b):=\{b^{\prime }\ge c:h_{\mathit{top}}(f{|}_{S_f^{+}(a,b^{\prime })})=h_{\mathit{top}}(f{|}_{S_f^{+}(a,b)})\}.\end{array}$$ Moreover, we obtain a sufficient and necessary condition for when \(E_{f}(a)=B_{f}(a)\) $E_f(a)=B_f(a)$ , extending the results of Baker and Kong (2020), Allaart and Kong (2023).