Consider matrix ODE:

$$

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

$$

Where $A(t)$ is a given periodic matrix with period $T$, i.e. $A(x)=A(x+T)$, for all $x\in \mathbb{R}$.

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

$$

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

$$

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

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

$$

B=\frac{1}{T}\log\left(\frac{\phi(T)}{\phi(0)}\right)

$$

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

In fact, matrices are not like numbers, the $\log$ function is much more complicated. We have:

$$

\log(A)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{A^n}{n} \quad \ldots(*)

$$

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

$$

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

$$

But this type of identity is only meaningful when $\left\lVert\frac{\phi(T)}{\phi(I)}\right\rVert<1$. So, is it true that for $\left\lVert\frac{\phi(T)}{\phi(I)}\right\rVert<1$ the equation is solved by $(*)$, and for $\left\lVert\frac{\phi(T)}{\phi(I)}\right\rVert\geq 1$ it does not have a solution?

The naive inspiration is wrong; the situation is similar to the $\mathbb{Q}_p$ case while $\log_p$ could extend to $D(p^{\frac{-1}{p-1}-})$ and the identity:

$$

\exp_p(\log_p(1+x))=1+x

$$

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

$$

\log[(1+X)(1+Y)]=\log(1+X+Y+XY)

$$

even when $\left\lVert X+Y+XY\right\rVert\geq 1$ and this process could be continued to the whole matrix space, and the identity enjoys the associated principle so $\log(X)$ is well-defined for all $X\in M_{2\times 2}$.

Now it is time to consider the rotation number, which is defined by:

$$

\lim_{n\to \infty}\frac{f^{n}(x)-x}{n}

$$

for $f:\mathbb{R}\to \mathbb{R}$ is a continuous increasing function.

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

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

$$

\phi : \mathbb{R} \to \mathbb{R}

$$

such that $\phi(x+1)=\phi(x)+1$.

Form the map:

$$

h:=\frac{1}{q} \sum _{n=1} ^q (\phi^{\circ n}-pn)

$$

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

Maslov index of a holomorphic disk

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

$$

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

$$

$$

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

$$

Proof sketch:

- $B=\frac{1}{T}\log\left(e^{\int_0^T A+f(t)dt}\right)= \frac{1}{T}\left(\int_{0}^T A+f(t)dt\right)$.
- The dynamic system is defined by: $W: \mathbb{R}^2-{0} \to \mathbb{R}^2-{0}$, $W( x)=B x$.
- This dynamic system $(\mathbb{R}^2-{0},W)$ is conjugate to the dynamic system $T:S_1\to S_1$. Not difficult to prove it is a homomorphism on $S_1$ and it is zero entropy by Pesin’s formula.

If $T: S_1\to S_1$ could be lifted to $\hat T:\mathbb{R} \to \mathbb{R}$, the rotation number is defined as:

$$

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

$$

This problem is not a good problem due to the philosophy, i.e. using the 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 $g$.