Arnold Conjecture & Floer Theory

Fixed points of Hamiltonian diffeomorphisms on the torus — Morse-Floer gradient flow lines connecting critical points

Mode (m,n): 1,1 Amplitude A: 1.00 Second harmonic B: 0.30 Flow time T: 2.00

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H = A·cos(2πmx)cos(2πny)
+ B·cos(4πx+4πy)
∇H = (-∂H/∂y, ∂H/∂x)
Fixed pts: ∇H = 0 mod ℤ²

Fixed points found: —

Arnold bound: 4mn = —

Arnold conjecture (proved by Floer 1989):
#fixed pts ≥ sum of Betti numbers = 4