Anosov diffeomorphisms are fascinating examples from dynamical systems. On the one hand, they exhibit great instability: if you take two nearby points in a space and repeatedly apply an Anosov diffeomorphism to that space, the points typically follow extremely different trajectories. On the other hand, Anosov diffeomorphisms also exhibit remarkable stability: if f is an Anosov diffeomorphism and g is a small perturbation of f, then (it is a theorem that) g and f are essentially the same (precisely, conjugate).

The study of these maps and their generalizations is the modern field of hyperbolic dynamics. I'll introduce you to lots of examples and some proof-by-picture of their remarkable properties.