# Is the set of conditions for Devaney's definition of chaos minimal?

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

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.

-

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

over a metric space see the following

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

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9223995804786682, "perplexity_flag": "head"}