Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am reading Devaney's definition of chaos. Which says:
Let $V$ be a set. $f:V \rightarrow V$ is said to be chaotic on V if

  1. $f$ has sensitive dependence on initial conditions
  2. $f$ is topologically transitive
  3. periodic points are dense in $V$

It seems that conditions 2 and 3 implies 1. Also, condition 2 seems to imply 1. Am I missing something here?

share|cite|improve this question

2 Answers 2

up vote 5 down vote accepted

Brooks, Cairns, Davis and Stacey proved that 2. and 3. imply 1. on metric spaces with an infinite number of points. There are no more redundancies in general metric spaces. However, if $V=[a,b]\subset\mathbb{R}$ and $f$ is continuous, then 2. implies 1. and 3. This is a result of Vellekop and Berglund, published in the American Mathematical Monthly, vol. 101, 1994.

share|cite|improve this answer
Thanks for the reply. However, I have some more related doubts. I would post them as a separate question so that it's easier to lookup later. – Abhishek Chanda Apr 26 '12 at 18:23
The link is not working, can you fix it? Thank you – lanzariel Oct 20 at 16:46
It used to work. I have updated the link. – Julián Aguirre Oct 20 at 18:44

over a metric space see the following

share|cite|improve this answer
The link is not working, can you fix it? Thank you. – lanzariel Oct 20 at 16:46

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.