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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?

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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.

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  • $\begingroup$ 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
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    $\begingroup$ It used to work. I have updated the link. $\endgroup$ – Julián Aguirre Oct 20 '15 at 18:44
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over a metric space see the following

http://www2.math.uu.se/~warwick/vt04/DynSyst/reading/BanksEtAl.pdf

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

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