Topological Entropy & Smale's Horseshoe
Smale's horseshoe map stretches a square, folds it, and maps it back. The invariant set is a Cantor set with topological entropy h=ln 2. Each iterate doubles the number of period-n orbits.
Left: the unit square after n applications of the horseshoe map. Vertical strips (preimages of the square) colored by symbolic itinerary — binary sequences. Right: orbit count (period-n points) grows as 2^n, confirming h_top = ln 2. The intersection of forward and backward iterates forms the Cantor invariant set.