I'm trying to find out the motivations that led V. Arnold to formulate his famous conjecture (I guess theorem by now) in the following form:
Let $(M,\omega)$ be a closed symplectic manifold (add whatever monotonicity condition here to avoid the need of virtual methods) and let $\phi$ be an Hamiltonian diffeomorphism, then the number of fixed points of $\phi$ is bounded below by the sum of the Betti numbers of $M$
Now Hamiltonian diffeomorphisms correspond to canonical transformations in physics and give rise to a family of Hamiltonians $\{H_t\}_{t \in [0,1]}$.
Q1 Why do we require in the standard statement of the weak Arnold conjecture, that the family of Hamiltonians is (say) $1$-periodic? Is it due to some interesting physic application?
In order to set up the action functional on the loop space of $M$ (the approach of Conley, Zehnder and Floer for example), we have to restrict to contractible periodic orbits $x(t)$ solutions of $$ \dot{x(t)}=X_{H_t}(x(t))$$ where $X_{H_t}$ is the Hamiltonian vector field associated with $H_t$.
Q2 Since there is a 1-1 bijection between fixed points of $\phi=\varphi^1_{X_{H_t}}$ and the periodic orbits defined above, does it mean that I'm potentially not counting certain fixed points which arises as NON-contractible periodic orbits?
Thanks a lot for any clarifications
