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Computing explicitly topological sequence entropy: the unimodal case

Victor Jiménez LópezJose Salvador Cánovas Peña — 2002

Annales de l’institut Fourier

Let W ( I ) denote the family of continuous maps f from an interval I = [ a , b ] into itself such that (1) f ( a ) = f ( b ) { 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 ( f ) of any map f W ( I ) respect to the sequence D = ( 2 m - 1 ) m = 1 .

Commutativity and non-commutativity of topological sequence entropy

Francisco BalibreaJose Salvador Cánovas PeñaVíctor Jiménez López — 1999

Annales de l'institut Fourier

In this paper we study the commutativity property for topological sequence entropy. We prove that if X is a compact metric space and f , g : X X are continuous maps then h A ( f g ) = h A ( g f ) for every increasing sequence A if X = [ 0 , 1 ] , and construct a counterexample for the general case. In the interim, we also show that the equality h A ( f ) = h A ( f | n 0 f n ( X ) ) is true if X = [ 0 , 1 ] but does not necessarily hold if X is an arbitrary compact metric space.

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