Chaos on the interval
a survey of relationship between the various kinds of chaos for continuous interval maps
Dynamical systems on the interval were widely studied because they are
among the simplest systems and nevertheless they turn out to have
complex dynamics. Many works on chaos were inspired by the behaviour of
interval maps. However these systems have many properties that
are not found on other spaces. As a consequence, one-dimensional
dynamics is very rich and worth a separate study.
The aim of this book is to survey the relations between the
various sorts of chaos and related notions for continuous interval
maps. The papers on this topic are numerous but very
scattered in the literature, sometimes little known or difficult to
find; some were originally published only in Russian or without proof.
The diagram below names the
main notions we will consider and sums up the links between them.
Chaos on the interval.
Volume 67 of University Lecture
Series, AMS, 2017.
pdf version (last revision April 12, 2018):
Errata and changes between the published book and the pdf version
Since there are additional citations, this shifts the numbers in the
- Remark 5.12: additional citation [Li, Moothathu, Oprocha] (arxiv v4).
- Lemma 5.43 is removed (and replaced by Remark 5.43);
Lemma 5.44 has a more precise statement and its proof is simpler; additional citation [Mai, Sun]
- Before Proposition 5.45: a few additional lines about the non
optimality of the bound in Proposition 5.45; additional citation
[Engelking] (arxiv v5).
- Erratum in Lemma 5.52 : the beginning of the proof was not well written.
It has been modified in the pdf version (arxiv v5).