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∈[0,1]:f(b)≤ fn(x) ≤ f(a)\ ∀ n≥0\, where the hole (a,b)⊂eq [0,1] satisfies a≤ c ≤ b and a≠ b. Let a∈[0,c] be fixed, we mainly focus on the following two bifurcation sets: Ef(a):=\b∈[c,1]:S+f(a,ε)≠ S+f(a,b) \ ∀ \ ε>b\, \ \ and Bf(a):=\b∈[c,1]:htop(S+f(a,ε))≠ htop(S+f(a,b)) \ ∀ \ ε>b\. By combinatorial renormalization tools, we give a complete characterization of the maximal plateau P(b) such that for all ε∈ P(b), htop(S+f(a,ε))=htop(S+f(a,b)). Moreover, we obtain a sufficient and necessary condition for Ef(a)=Bf(a), which partially extends the results in allaart2023 and baker2020.
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