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

• Yun Sun ^[1] ; Bing Li ^[1]
  1. [1] South China University of Technology

     South China University of Technology

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 1, 2025
• Idioma: inglés
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• Resumen

  • Consider an expansive Lorenz map f defined on I = [0, 1] and let c be its discontinuity. The survivor set under consideration is represented as S+ f (a, b) := {x ∈ I : f (b) ≤ f n(x) ≤ f (a), ∀ n ≥ 0}, where (a, b) ⊆ I satisfies a ≤ c ≤ b and a = b. We mainly study the following two bifurcation sets related to survivor set and its topological entropy htop:

    E f (a) := {b ≥ c : S+ f (a, ) = S+ f (a, b), ∀ > b}, and B f (a) := {b ≥ c : htop( f |S+ f (a,)) = htop( f |S+ f (a,b)), ∀ > b}.

    By utilizing combinatorial renormalization techniques, we give the kneading sequence s of the endpoints of the platform P(b) := {b ≥ c : htop( f |S+ f (a,b )) = htop( f |S+ f (a,b))}.

    Moreover, we obtain a sufficient and necessary condition for when E f (a) = B f (a), extending the results of Baker and Kong (2020), Allaart and Kong (2023).