# Eloquent theory

Consider matrix ODE:

$latex \dot{\phi}(t)=A(t)\phi(t)$

Where $latex A(t)$ is a given periodic matrix with period $latex T$, i.e. $latex A(x)=A(x+T), \forall x\in R$.

Then the solution $\phi(t)$ satisfied identity:

$latex \phi(t+T)=\phi(t)\phi^{-1}(0)\phi(T)$.

This could be explained as $latex \phi^{-1}\phi(T)=\int_{0}^T\phi(t)$.

Now we consider to solve the equation: $latex e^{TB}=\phi^{-1}(0)\phi(T)$. At least formally it could be solved:

$latex B=\frac{1}{T}log(\frac{\phi(T)}{\phi(0)})$.

(Unfortunately $latex log$ is a multi-value function so $latex B=B_0+2\pi ik I$, where $latex I$ is the identity matrix and $latex B$  is a solution of $latex e^{TB}=\phi^{-1}(0)\phi(T)$.) This argument is false.

In fact matrix is not like numbers, the $latex log$ function is much more complicated. we have,

$latex log(A)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{A^n}{n} …(*)$

So to solve $latex e^{TB}=\frac{1}{T}(\frac{\phi(T)}{\phi(I)})$, it is equivalent to :

$latex B=\frac{1}{T}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}(\frac{\phi(T)}{\phi(I)})^n}{n}$

But this type of identity only meaningful when $latex ||\frac{\phi(T)}{\phi(I)}||<1$, so is it true that for $latex ||\frac{\phi(T)}{\phi(I)}||<1$ the equation is solved by $latex (*)$, and for $latex ||\frac{\phi(T)}{\phi(I)}||\geq 1$ it do not have solution?

The naive inspirit is wrong, the situation is similar to the $latex \mathbb Q_p$ case while $latex log_p$ could extend to $latex D(p^{\frac{-1}{p-1}-})$ and the identity

$latex exp_p(log_p(1+x))=1+x$

always holds for $latex x\in D(p^{\frac{-1}{p-1}-})$. The key observation is $latex log[(1+Y)(1+Y)]=log(1+X)+log(1+Y)$ always holds when $latex ||X||,||Y||<1$, this will lead to a reasonable value of

$latex log[(1+X)(1+Y)]=log(1+X+Y+XY)$

even when $latex ||X+Y+XY||\geq 1$ and this process could be continue to the whole matrix space and the identity enjoy the accosted principle so $latex log(X)$ is well-defined for all $latex X\in M_{2\times 2}$.

Now it is time to consider the rotation number, which is defined by $latex \lim_{n\to \infty}\frac{f^{n}(x)-x}{n}$ for $latex f:R\to R$ is a continuous increasing function.

And I do not know how to associated a dynamic system for the matrix $latex B$ given here, but in any case it seems iff it is given by a hemoermorphifm then the rotation number is zero due to the following reason:

Consider $latex \mathbb S^1$ as the quotient $latex \mathbb R/\mathbb Z$. Your homeomorphism $latex f$ lifts to a homeomorphism

$latex \phi : \mathbb R \to \mathbb R$ such that $latex \phi(x+1)=\phi(x)+1$.

Form the map $latex h:=\frac{1}{q} \sum _{n=1} ^q (\phi^{\circ n}-pn)$, where $latex \phi ^{\circ n}$ is the composition $latex n$ times of $latex \phi$ with itself. By construction $latex h\circ \phi = h+\frac{p}{q}$ and $latex h(x+1)=1+h(x)$, so that $latex h$ factors as a homeomorphism of the circle conjugating $latex f$ to the rotation.
By the way this approach wors in $latex \mathbb R^n$ too.

Maslov index of a holomorphic disk

A natural way to understand the rotation number here is according the way of maslov index, we have the following formula:

$latex f(A)=\int_{\Gamma}\frac{1}{2\pi i}\frac{f(\lambda)}{\lambda I-A}f(\lambda)d\lambda$

$latex TB=log(\frac{\phi(T)}{\phi(0)})=log(\int_0^T \phi'(\lambda)d\lambda)=\int_0^T log(\phi'(\lambda))d\lambda=\int_0^T log(A+F(t))d\lambda$

Proof sketch:

1.$latex B=\frac{1}{T}log(e^{\int_0^T A+f(t)dt})= \frac{1}{T}(\int_{0}^T A+f(t)dt)$.

2. The dynamic system is defined by : $latex W: R^2-\{0\} \to R^2-\{0\}$, $latex W( x)=B x$.

3. this dynamic system $latex (R^2-\{0\},W)$ is conjugate to the dynamic system $latex T:S_1\to S_1$,  Not difficult to proof it is a homomorphism on $latex S_1$ and it is zero entropy by Pesin’s formula

If $latex T: S_1\to S_1$ could lifting to $\hat T:R \to R$ the rotation number is defined as :

$latex \lim_{n\to \infty}\frac{\hat T^n(x)}{n}$

This problem is not a good problem due to the philosophy, i.e. use rotation number to describe the information of a hamiltonian flow is not satisfied, in fact it is difficult to establish a suitable definition of “rotation number”! But this is the first crucial thing to establish a theorem!

Hamiltonian flow

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. A Hamiltonian vector field is a geometric manifestation of Hamilton’s equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.

Hamiltonian vector fields can be defined more generally on an arbitrary Poisson manifold. The Lie bracket of two Hamiltonian vector fields corresponding to functions f and g on the manifold is itself a Hamiltonian vector field, with the Hamiltonian given by the Poisson bracket of f and g.

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