American Institute of Mathematical Sciences

October  2012, 32(10): 3587-3620. doi: 10.3934/dcds.2012.32.3587

Transport, flux and growth of homoclinic Floer homology

 1 Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, United States

Received  February 2011 Revised  April 2012 Published  May 2012

We point out an interesting relation between transport in Hamiltonian dynamics and Floer homology. We generalize homoclinic Floer homology from $\mathbb{R}^2$ and closed surfaces to two-dimensional cylinders. The relative symplectic action of two homoclinic points is identified with the flux through a turnstile (as defined in MacKay & Meiss & Percival [19]) and Mather's [20] difference in action $\Delta W$. The Floer boundary operator is shown to annihilate turnstiles and we prove that the rank of certain filtered homology groups and the flux grow linearly with the number of iterations of the underlying symplectomorphism.
Citation: Sonja Hohloch. Transport, flux and growth of homoclinic Floer homology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3587-3620. doi: 10.3934/dcds.2012.32.3587
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