Topological dynamics is a subfield of the area of dynamical systems. The main focus is properties of dynamical systems that can be formulated using topological objects.
Classical objects of study in topological dynamics are (iterates of) homeomorphisms (or continuous self-maps) of compact metric spaces, or continuous flows (or semiflows) of compact metric spaces. That is, topological dynamics is interested in group homomorphisms $\alpha_\bullet:T\to \operatorname{Homeo}(X)$ with $\alpha:T\times X\to X, (t,x)\mapsto \alpha_t(x)$ continuous, where $T=\mathbb{Z}$ or $T=\mathbb{R}$ and $X$ is a compact metric space. Topologico-dynamical objects that can be attached to such a dynamical system include $\alpha-$ and $\omega-$limit sets, nonwandering sets, chain recurrent sets, various sets of return time and topological entropy. By using various sets of return time one can formulate different mixing properties in the topological category in a way analogous to the mixing properties in ergodic theory.
More generally one can consider group actions of more general (typically locally compact Hausdorff second countable) groups on a topological space (typically Hausdorff, but not necessarily compact, nor necessarily metrizable). Topological entropy is a uniformity property; accordingly actions of groups on uniform spaces via uniform homeomorphisms are also considered.
In the classical case a theorem by Krylov-Bogoliubov guarantees that a topological dynamical system has at least one invariant Borel probability measure. In this way there is an intimate relation between topological dynamics and ergodic theory. Indeed one may consider the Krylov-Bogoliubov theory as attaching to a topological dynamical system another topological dynamical system, where the state space is the compact metric space of probability measures on the original state space.
