Milnor-Thurston 1988: The kneading invariant K(r) completely classifies the topological dynamics. Two maps are conjugate iff they have the same kneading sequence. Period-n orbits correspond to sequences with K^n = K.
Map with partition and orbit
Bifurcation diagram (colored by itinerary)
Kneading: —
Period: —
Topological entropy: —
Itinerary of x₀ (L=left of critical point, R=right):
—
Symbol sequence n-gram frequency (grammar complexity)
Symbolic dynamics replaces continuous analysis with combinatorics. The kneading invariant K encodes the itinerary of the critical point — the most important orbit.