# Counting fixed points for Hamiltonian symplectomorphisms on $T^{2}$

This question is motivated by the Lorenz curve used in economic analysis and also the Penrose diagram used in general relativity, used by physicists in order to visualise causal relationships in compactified Minkowski space time models.

It is also motivated deeply by Hamiltonian mechanics, symplectic geometry, and contact geometry (which can be viewed as the odd dimensional counterpart of symplectic geometry). Hamiltonian mechanics was a reformulation of Newtonian mechanics. In Hamiltonian mechanics one studies phase spaces of physical systems, symplectic flows and many other topics.

From a mathematical standpoint, this question is motivated from modern trends in differential geometry, topology, abstract algebra and mathematical physics.

Consider the families of planar curves for $$x,y\in \Re(0,1)$$ and $$\Re(s) \ge1:$$

$$\zeta:= \{ (x, y) \in \Bbb R^2 | x^s + y^s = 1 \}$$ $$\tau:= \{ (x, y) \in \Bbb R^2 | (1-x)^s + y^s = 1 \}$$ $$\psi:= \{ (x, y) \in \Bbb R^2 | x^s + (1-y)^s = 1 \}$$ $$\phi:= \{ (x, y) \in \Bbb R^2 | (1-x)^s + (1-y)^s = 1 \}.$$

Interpreting $$\zeta,\tau,\psi,$$ and $$\phi$$ as phase spaces allows the interpretation of them as infinite dimensional manifolds, specifically infinite dimensional symplectic manifolds:$$(\zeta,\omega),(\tau,\omega),(\psi,\omega),(\phi,\omega).$$

What follows is a natural Hamiltonian vector field, which defines a Hamiltonian flow on each of the symplectic manifolds.

The process of lifting these manifolds into $$\Bbb R^3$$ can be achieved with homotopic maps. See the answer by Paul Frost here: https://math.stackexchange.com/questions/2895816/existence-of-homotopic-map. The key thing to understand is that each of the four symplectic manifolds are unique projections of the curves lifted via homotopy. In other words, one can project the lifted curves onto the planar curves bijectively. The image provides the intuition for what a single lift looks like. But in reality there are infinitely many of these lifts each at different heights above the planar curves.

Question:

Consider the symplectic manifolds defined above. I'm attempting to count the fixed points for Hamiltonian symplectomorphisms on $$T^2,$$ and generally $$T^{2n}.$$ What's the best way to attack this problem?

This is a shape I built of some geodesics to try to get a better understanding of the geometry of the object and the phase space of the particle ensemble for a specific configuration in three dimensions:

The image relates to the equations because each white strand in the image corresponds to a strand of a geodesic on a $$2$$-manifold, namely $$S^2.$$ Each white geodesic also corresponds to a specific planar curve listed above for a specific value of $$s$$. In the picture $$s=2.$$ The singularities can be viewed as invariants or fixed points because they don't change spatial location for the purpose of this question. Varying $$s$$ gauges the way the shape looks. As $$s$$ approaches infinity the shape will look like a cube. As $$s$$ tends to $$1,$$ the shape will look like $$2$$ pairs of perpendicular lines situated in $$3$$-space.

Maybe it helps to say that the image is the shape of the intersections of $$8$$ identical copies of $$S^2$$ with finitely many geodesics shown. It is like a higher dimensional venn diagram if you will.

• Your questions are kind of vague. Generally "pick a question" questions don't do very well here. If you could revise your post to have just one question, and make it more precise that would help. But the way your questions are written they would appear to have fairly simple and uninformative answers. – Ryan Budney May 28 at 14:26
• Okay I will do that within 2 hrs – Ultradark May 28 at 14:47