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Commutativity and non-commutativity of topological sequence entropy

Francisco Balibrea, Jose Salvador Cánovas Peña, Víctor 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.

Commuting contractive families

Luka Milićević (2015)

Fundamenta Mathematicae

A family f₁,..., fₙ of operators on a complete metric space X is called contractive if there exists a positive λ < 1 such that for any x,y in X we have d ( f i ( x ) , f i ( y ) ) λ d ( x , y ) for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin’s conjecture is true for three operators, provided that λ is sufficiently small.

Compact covering mappings and cofinal families of compact subsets of a Borel set

G. Debs, J. Saint Raymond (2001)

Fundamenta Mathematicae

Among other results we prove that the topological statement “Any compact covering mapping between two Π⁰₃ spaces is inductively perfect” is equivalent to the set-theoretical statement " α ω ω , ω L ( α ) < ω "; and that the statement “Any compact covering mapping between two coanalytic spaces is inductively perfect” is equivalent to “Analytic Determinacy”. We also prove that these statements are connected to some regularity properties of coanalytic cofinal sets in (X), the hyperspace of all compact subsets of a Borel...

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