The Number of Periodic Points of Surface Symplectic Diffeomorphisms

The paper’s Theorem 1.1 states exactly the solver’s claim and proves it via two ingredients: (i) a forcing result (Theorem 1.13) that if a contractible fixed point has nontrivial local Floer homology and nonzero mean index, then for all sufficiently large primes p there is a simple p-periodic orbit, relying on the support of Floer–Novikov homology on surfaces and an isolation-by-blow-up-and-gluing argument; and (ii) a dichotomy (Theorem 1.14) which, under finiteness of contractible fixed points and nontrivial flux, guarantees either such a point exists or there is a symplectically degenerate extremum that also forces simple prime-periodic orbits. The ‘moreover’ part is established by a doubling (blow-up-and-gluing) contradiction comparing the global Floer–Novikov rank to the sum of local ranks. These steps closely match the model’s outline and details, including the use of admissible iterations, mean-index growth, support windows, and the gluing construction. See Theorem 1.1 and its discussion; the proof outline for Theorem 1.13; the dichotomy Theorem 1.14; the local Floer support window; and the blow-up-and-gluing discussion used in both the forcing and ‘moreover’ arguments.