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

Abstract

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

18 pages

References in corpus (3)

• Intermediate $\beta$-shifts as greedy $\beta$-shifts with a hole
• Subshifts of finite type and matching for intermediate $\beta$-transformations
• Topological expansive Lorenz maps with a hole at critical point