Computing explicitly topological sequence entropy: the unimodal case

Jiménez López, Victor; Cánovas Peña, Jose Salvador

Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1093-1133.

Nous considérons $W\left(I\right)$ la famille de fonctions $f$ continues de l’intervalle $I=[a,b]$ sur lui–même, telles que (1) $f\left(a\right)=f\left(b\right)\in \{a,b\}$; (2) elles sont constituées de deux morceaux monotones; et (3) elles ont des points périodiques de périodes toutes les puissances de $2$ exactement. L’objectif principal de ce travail est de calculer explicitement l’entropie topologique séquentielle $h_D\left(f\right)$ de tout élément $f$ de $W\left(I\right)$ par rapport à la suite $D={\left(2^{m-1}\right)}_{m=1}^{\infty }$.

Let $W\left(I\right)$ denote the family of continuous maps $f$ from an interval $I=[a,b]$ into itself such that (1) $f\left(a\right)=f\left(b\right)\in \{a,b\}$; (2) they consist of two monotone pieces; and (3) they have periodic points of periods exactly all powers of $2$. The main aim of this paper is to compute explicitly the topological sequence entropy $h_D\left(f\right)$ of any map $f\in W\left(I\right)$ respect to the sequence $D={\left(2^{m-1}\right)}_{m=1}^{\infty }$.

Classification : 37B40, 26A18, 54H20
Keywords: map of type $2^\infty $, topological sequence entropy, unimodal map
Mot clés : fonction de type $2^\infty $, entropie séquentielle topologique, fonction unimodale

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