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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.04008 |
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| _version_ | 1866929610953850880 |
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| author | Sun, Yun Li, Bing |
| author_facet | Sun, Yun Li, Bing |
| 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04008 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two bifurcation sets of expansive Lorenz maps with a hole at the critical point Sun, Yun Li, Bing Dynamical Systems 37E05, 37B10 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}. |
| title | Two bifurcation sets of expansive Lorenz maps with a hole at the critical point |
| topic | Dynamical Systems 37E05, 37B10 |
| url | https://arxiv.org/abs/2404.04008 |