Thomas Ohlson Timoudas: Cantor sets, shift maps and chaos

In dynamics we study the long-term behaviour of orbits, namely the distribution of the sets Orb(x) = {x, f(x), f(f(x)), ...} of iterates for some specified (continuous) map f: X -> X.

Invariant sets (closed sets U satisfying f(U) = U) provide us a notion of subspace. I will show how Cantor sets sometimes naturally appear as invariant sets, even for very simple maps f.

The quadratic map f(x) = ax(1-x) will take center stage in this surprising drama, featuring the shift map in the supporting role.