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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.04008 |
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Table of Contents:
- 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: $$ E_{f}(a):=\{b\in[c,1]:S^{+}_{f}(a,ε)\neq S^{+}_{f}(a,b) \ \forall \ ε>b\}, \ \ {\rm and} $$ $$ B_{f}(a):=\{b\in[c,1]:h_{top}(S^+_{f}(a,ε))\neq h_{top}(S^+_{f}(a,b)) \ \forall \ ε>b\}. $$ By combinatorial renormalization tools, we give a complete characterization of the maximal plateau $P(b)$ such that for all $ε\in P(b)$, $h_{top}(S^+_{f}(a,ε))=h_{top}(S^+_{f}(a,b))$. 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}.