| 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. |
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