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?


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.

  • $\begingroup$ 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. $\endgroup$ – Abhishek Chanda Apr 26 '12 at 18:23
  • $\begingroup$ The link is not working, can you fix it? Thank you $\endgroup$ – ThePunisher Oct 20 '15 at 16:46
  • 1
    $\begingroup$ It used to work. I have updated the link. $\endgroup$ – Julián Aguirre Oct 20 '15 at 18:44

over a metric space see the following


  • $\begingroup$ The link is not working, can you fix it? Thank you. $\endgroup$ – ThePunisher Oct 20 '15 at 16:46

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