Understanding topological dynamics of hyperbolic dynamical systems via examples

Topological dynamics constitutes the study of asymptotic properties of orbits under flows or maps on the Hausdorff phase space. Hyperbolic dynamics is the study of differentiable flows or maps that are usually characterized by the presence of expanding and contracting directions for the associated derivative on some manifold. We study some topological dynamics, essentially the property of `proximality', of two prototype examples of hyperbolic dynamical systems - \emph{Arnold's Cat Map} and \emph{Smale's Horseshoe Map} as an attempt to find some analogies in these two directions of study.