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Axiomatic definition of the topological entropy on
the interval
Ll. Alseda?, S. Kolyada, J. Llibre and L?. Snoha
February 7, 2002
Abstract The aim of this paper is to give an axiomatic definition of the
topological entropy for continuous interval maps and, in such a way, to shed
some more light on the importance of the different properties of the topolog-
ical entropy in this setting. We give two closely related axiomatic definitions
of topological entropy and an axiomatic characterization of the topological
chaos.
1 Introduction
The topological entropy as a measure of the complexity of a continuous self-map
of a compact topological space was introduced by Adler, Konheim and McAndrew
[1] and has been studied by many authors. In particular, in compact metric spaces
an equivalent definition has been found by Bowen [8] and Dinaburg [10]. For the
definition and main properties of topological entropy we refer the reader to [2] and
[21] (the book [2] is particularly focused on interval maps and very often, when
we use well known facts, we refer the reader to this book instead to the original
paper).
The idea of giving an axiomatic definition of entropy belongs to Rohlin who gave
in [16] an axiomatic definition of the measure-theoretic entropy of an automorphism
of a Lebesgue space. Later an analogue of the Rohlin’s result for a Zd-action for
every d ≥ 2 was proved by Kamin?ski in [13]. An axiomatic definition of topological
entropy for endomorphisms on compact groups was found by Stojanov [20]. For
completeness we recall that there are also many papers dealing with axiomatic
characterizations of various kinds of entropy in the context of information theory.
2000 Mathematics Subject Classification: Primary 37B40, 37E05; Secondary 26A18, 54H20.
Key words and phrases: interval maps, topological entropy, axiomatic definition, chaos, iterate,
iterative root, factor, lower semicontinuity, pouring water, semiconjugacy.
1
Our main concern in this paper is to give an axiomatic def
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