TITLE: The relationship between low dimensional topology and dynamics
QUESTION [7 upvotes]: I am just curious how dynamics get connected with low dimensional topology. Or it is just that we have now powerful computing machines therefore it is natural to use them on topological problems. What kind of problems researchers are interested in working at the intersection of dynamics and low dimensional topology. 
Any suggestion for reading is greatly appreciated. The phrase sound fascinating, but have almost no understanding. Thanks.

REPLY [6 votes]: The machinery of Markov partitions and stable/unstable foliations for Anosov and Axiom A diffeomorphisms was adapted to several different bits of low dimensional topology. In the Nielsen-Thurston classification of surface mapping classes alluded to by others, Markov partitions are the machinery behind the construction and study of pseudo-Anosov homeomorphisms, and the stable/unstable foliations are very explicit in that study. In the Bestvina-Feighn-Handel classification of outer automorphisms of free groups, the concepts of Markov partitions lie behind the theory of train tracks and relative train tracks, and the lamination part of the theory comes through quite explicitly in the theory of attracting laminations for outer automorphisms.