SOME RESULTS ON ENTROPY AND SEQUENCE ENTROPY

Author:

BALIBREA F.1,LÓPEZ V. JIMÉNEZ1,PEÑA J. S. CÁNOVAS2

Affiliation:

1. Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, Apto. de Correos 4021, 30100 Murcia, Spain

2. Departamento de Matemática Aplicada, Escuela Politécnica de Cartagena, Paseo de Alfonso XIII, 34–36, 30203 Cartagena, Spain

Abstract

In this paper we study some formulas involving metric and topological entropy and sequence entropy. We summarize some classical formulas satisfied by metric and topological entropy and ask the question whether the same or similar results hold for sequence entropy. In general the answer is negative; still some questions involving these formulas remain open. We make a special emphasis on the commutativity formula for topological entropy h(f ◦ g)=h(g ◦ f) recently proved by Kolyada and Snoha. We give a new elementary proof and use similar ideas to prove commutativity formulas for metric entropy and other topological invariants. Finally we prove a Misiurewicz–Szlenk type inequality for topological sequence entropy for piecewise monotone maps on the interval I=[0, 1]. For this purpose we introduce the notion of upper entropy.

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Modelling and Simulation,Engineering (miscellaneous)

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