Title: Computational complexity, torsion-freeness of homoclinic Floer homology, and homoclinic Morse inequalities,
Abstract: Floer theory was originally devised to estimate the number of 1-periodic
orbits of Hamiltonian systems. In earlier works, we constructed Floer homology
for homoclinic orbits on two dimensional manifolds using combinatorial
techniques. In the present paper, we study theoretic aspects of computational
complexity of homoclinic Floer homology. More precisely, for finding the
homoclinic points and immersions that generate the homology and its boundary
operator, we establish sharp upper bounds in terms of iterations of the
underlying symplectomorphism. This prepares the ground for future numerical
works.
Although originally aimed at numerics, the above bounds provide also purely
algebraic applications, namely
1) Torsion-freeness of primary homoclinic Floer homology.
2) Morse type inequalities for primary homoclinic orbits.