Are 'topological transformation groups' still studied in current research? I was looking at the book by Gottschalk and Hedlund on Topological Dynamics; it uses topological transformation groups in the beginning and terms like Unimorphism, which seem obscure in current literature. Is Topological Dynamics in general not an active area of research now?
 A: Topological dynamics is a very broad term with several very active sub disciplines or closely related areas (symbolic dynamics, homogeneous dynamics, continua theory, foliation theory, ergodic theory, etc.).
It may be that topological transformation groups are not currently an active area of research (though they may be, I don't know the literature well enough. [Edit] see Moishe Cohen's answer), but topological dynamics as a discipline is active and thriving, with regular well-attended international conferences. The Spring Topology and Dynamical Systems Conference and the Summer Conference on Topology and Its Applications come to mind as always having a particular focus on dynamics, as well as many others.
Perhaps things have shifted both in nomenclature and focus since the 1950s, but the activity doesn't seem to have died down much.
A: People are still working on topological transformation groups, although the area is not as active as it was in 1940s-1960s. One of the major open problems in this area is the Hilbert-Smith Conjecture, which states that if a locally compact topological group $G$ acts faithfully on a topological manifold $M$, then $G$ is a Lie group. Recently 
John Pardon proved this conjecture for 3-dimensional manifolds: 
J. Pardon, The Hilbert-Smith conjecture for three-manifolds. 
J. Amer. Math. Soc. 26 (2013), no. 3, 879–899.  
