# Computational complexity, torsion-freeness of homoclinic Floer homology, and homoclinic Morse inequalities

## TL;DR

This paper investigates the computational complexity of homoclinic Floer homology, establishing bounds that lead to algebraic results like torsion-freeness and Morse inequalities for homoclinic orbits.

## Contribution

It provides sharp upper bounds on the complexity of computing homoclinic Floer homology, enabling algebraic insights such as torsion-freeness and Morse inequalities.

## Key findings

- Sharp upper bounds for homoclinic points and immersions
- Proof of torsion-freeness of primary homoclinic Floer homology
- Establishment of Morse-type inequalities for homoclinic orbits

## 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.