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.
A very recent example:
The Kahn-Markovic result that a closed hyperbolic three manifold has an essential immersed hyperbolic Riemann surface, i.e., the map on fundamental groups is injective. This is a solution to a long-standing open problem (and in particular got them the 2012 Clay research award).
The proof is heavily based on ergodic theory, in particular on the mixing property of the geodesic flow.
One important concept at the intersection of topology and dynamics is the "Sullivan dictionary": a fairly direct correspondence between Kleinian groups and complex dynamics that, in certain situations, allows one to translate results (and even, to some extent, proofs) in the Kleinian group setting to the rational maps setting, and vice versa. McMullen has a nice exposition on this at the beginning of "Renormalization and 3-manifolds which fiber over the circle" (you can see a table of some parts of this "dictionary" in the Google reader). Also, Canary has a few sets of slides online about this correspondence (see Kleinian groups and the Sullivan dictionary I,II,III). I've heard that some of the recent big results about Kleinian groups have analogues in the rational maps setting, but that those "conjectures" are open.
Also, it seems like Peter Jones has done some work on extending this dictionary to SLE (see online slides "Some remarks on SLE and an extended Sullivan dictionary"). Of course, all of these correspondences are unique to the hyperbolic setting.
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.
Robert Ghrist started out his (prolific) research career investigating knotted trajectories in dynamical systems. http://www.math.upenn.edu/~ghrist/preprints-dynamics.html
Konstantin Mischaikow has been working on topological classification of dynamical behaviours. http://www.math.rutgers.edu/~mischaik/pub_page/paperlist_dynamics.html
One thing I have heard people show interest in is a decomposition of a low-dimensional dynamic system into cells with homogenous behaviour -- much in the vein of Morse theory: you pick out attractive and repulsive submanifolds, and regions of homogenous flow (starting out and ending up in the same sources and sinks), and this gives you (hopefully) a CW complex describing the dynamics.
There probably is more to it, but these are the things I have heard of.
I have an answer in a different direction. It has been developed a theory (starting with Mañe, and then Bonatti-Diaz-Pujals-Ures) showing that robust dynamical behaviour (in the $C^1$-topology) implies the existence of $Df$-invariant geometric structures (hyperbolicity, partial hyperbolicity, dominated splitting...). In particular, one knows that a $C^1$-robustly transitive diffeomorphism is:
1) (Mañe) Anosov if $dim M=2$. By Franks' results, this implies that $M= T^2$ and $f$ is isotopic to a linear Anosov automorphism.
2) (Diaz-Pujals-Ures) Partially hyperbolic if $dim M=3$.
3) (Bonatti-Diaz-Pujals) Admits a dominated splitting if $dim M \geq 3$.
See the book by Bonatti-Diaz-Viana or the following survey (http://arxiv.org/abs/0912.2896 in french) for more information.
It seems natural to expect that the existence of such $Df$-invariant geometric structures impose some restrictions on the topology of the manifold which admits such diffeomorphisms. So, a relationship between $C^1$-robust dynamical behaviour and topological properties. It seems natural to study these questions in low-dimensions (such as in 1) the question is "solved" in dimension 2). In dimension 3, there are some guesses on which are the manifolds supporting (strong) partially hyperbolic systems. See the work of Bonatti-Wilkinson and Brin-Burago-Ivanov (they are in the author's websites). See also http://www.mat.puc-rio.br/edai/textos/potrie.pdf for some related discussion.
In particular, just to mention an open question which I find really interesting: Does there exists a robustly transitive diffeomorphism on $S^3$? Partial negative answers were given by Diaz-Pujals-Ures and Burago-Ivanov, but in all its generality it remains open.
Expanding on jc's comment (in particular the reference to Boyland's survey).
One way in which dynamics and low dimensional topology get mixed is via the Nielsen-Thurston classification. The point is that this classification allows one to more or less single out a `simplest' homeomorphism in each isotopy class. Knowing this has allowed people to show that simple topological patterns (usually involving only a finite number of points) can imply very complicated dynamics.
A striking examples of this is given by Llibre and MacKay's theorem for torus homeomorphisms (the existence of 3 non-collinear rotation vectors implies positive entropy).
A nice exposition with several examples is given by Mackay here.
