|For special classes of functions, Devaney's three attributes of chaos are not independent. For example,|
|Banks, et.al. showed that|
|*   if f:X → X is continuous and distances can be measured on X,|
|*   then sensitivity to initial conditions is a consequence of Devaney's other two conditions.|
|Vellekoop and Berglund showed that|
|*   for continuous functions taking an interval to itself,|
|*   the mixing condition (also called transitivity) implies the other two of Devaney's requirements.|
Return to Mathematical Definition of Chaos.