3
$\begingroup$

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?

2 Answers 2

6

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.

  • 1
    It used to work. I have updated the link.2015-10-20
0

over a metric space see the following

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

  • 0
    The link is not working, can you fix it? Thank you.2015-10-20