the invariant set of the smale horseshoe is obtained by intersecting all the horizontal & vertical strips in the square under iterations of the map & its inverse. this type of set is a cantor set. it looks like dust, but is uncountably infinite.
twitter is going to compress this into mush, i'm sure. there is a better quality version up on youtube:
what do you do with this cantor set? you prove that the horseshoe map on the invariant is "the same as" (topologically conjugate to) the shift map on the space of bi-infinite strings of 0's and 1's.
from this, one easily deduces the existence & denseness of periodic orbits, sensitive dependence on initial conditions, the existence of a dense orbit, and so much more... symbolic dynamics is one of my favorite topics in dynamical systems.
