Complicated dynamics from simple topological hypotheses

There are many types of dynamical system for which quite simple topological hy potheses imply very complicated behaviour. A well-known example is 'period 3 implies chaos' for continuous maps of an interval. A diverse range of more sophisticated examples has been produced by; many researchers and is reviewed here. The relevance to physical applications - like mixing in fluid flows. chaotic motion of mechanical linkages, and fast dynamos - is explored.