Journal of Symplectic Geometry

Volume 11 (2013)

Number 3

Action–index relations for perfect Hamiltonian diffeomorphisms

Pages: 449 – 474



Mike Chance (Department of Mathematics,Vanderbilt University, Nashville, Tennessee, U.S.A.)

Viktor Ginzburg (Department of Mathematics, University of California at Santa Cruz)

Basak Gürel (Department of Mathematics, Vanderbilt University, Nashville, Tennessee, U.S.A.)


We show that the actions and indexes of fixed points of a Hamiltonian diffeomorphism with finitely many periodic points must satisfy certain relations, provided that the quantum cohomology of the ambient manifold meets an algebraic requirement satisfied for projective spaces, Grassmannians and many other manifolds. We also refine a previous result on the Conley conjecture for negative monotone symplectic manifolds, due to the second and third authors, and show that a Hamiltonian diffeomorphism of such a manifold must have simple periodic orbits of arbitrarily large period whenever its fixed points are isolated.

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