ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 58,54 Hiroe Oka, Ryukoku University, Japan Connectivity of iso-topological entropy curve for symmetric PL bimodal maps We study the topological entropy of a two parameter family of symmetric piecewise-linear bimodal maps F_{λ,μ} on [−1, 1], which is given by

F_{λ,μ}(x) = \{ \begin{matrix}λ(x − 1/μ) − 1 & x ∈ I_{R} \\ −μx & x ∈ I_{M} \\ λ(x + 1/μ) + 1 & x ∈ I_{L}, \end{matrix}  (1)

where μ > 1, λ > 0 and I_{L} = [−1, −1/μ], I_{M} = [−1/μ, 1/μ], I_{R} = [1/μ, 1]. Let h(λ, μ) be the topological entropy for F_{λ,μ}. The main result is: Theorem: Given a constant c with 0 < c < log 3, the iso-entropy curve given by h(λ, μ) = c is connected. We also obtain the monotonicity of h(λ, μ) with respect to the parameter μ. For the monotonicity of the parameter λ, we have to restrict the parameter region. Some numeric simulations are shown. This is joint work with S. Nakamura. Reference: M. Misiurewicz and E. Visinescu, Kneading sequences of skew tent maps, Ann. Inst. H. Poincaré, 27 (1991), 125–140.