Non-twist maps of the annulus and their lack of fixed points

Question

I have been wondering about the existence of a kind of `counterexample' to a modification of the Poincare-Birkhoff theorem which badly breaks the twist condition.

Let me state a variant of the Poincare-Birkhoff theorem in a way that is convenient here:

Let $f$ be a symplectomorphism of $S^1 \times [0,1]$. Choose a lift $F$ of $f$ to $\mathbb{R} \times [0,1]$, and write $F_i = F|_{\mathbb{R} \times \{i\}}$ and $f_i = f|_{S^1 \times \{i\}}$ for $i=0,1$. Then we have well defined rotation numbers $r_{F_0}$ and $r_{F_1}$. Suppose they are distinct and that $f_0$, $f_1$ are conjugate to rotations; then $f$ has a pair of $q$-periodic points for every rational number $p/q$ written in lowest terms satisfying $r_{F_0} \leq p/q \leq r_{F_1}$.

(My apologies if I misstated this!)

Of course, usually the Poincare-Birkhoff theorem is stated in terms of the twist condition at the boundary of $f$, and for homeomorphisms. Rotations certainly satsify the twist condition.

A natural way to badly violate the twist condition is to ask for $F_0$ to have rotation number zero and to have nondegenerate fixed points at the boundary. In that case, $f_0$ has an equal number of attracting and repelling fixed points interspersed with one another.

My question is: suppose that $f_1$ is still a rotation, say an irrational one. What can be said about the fixed points of f that do not lie on $S^1 \times \{0\}$? I would like to focus on the case where $f$ is smooth. For certain $f_0$ which have degenerate fixed points (say, with a single degenerate fixed point), we can arrange for $f$ to have no fixed points so long as the rotation number of $f_1$ is small, say less than $1/10$. (In fact, this can be accomplished as the flow of a time-independent hamiltonian.) Thus, the natural question what happens if we require that $f_0$ has nondegenerate fixed points.

In that case, I am curious: can one find an $f$ with $f_1$ a small irrational rotaion such that $f$ has no fixed points except for the fixed points of $f_0$, which we assume is a nondegenerate circle diffeomorphism of rotation number zero?

The natural extensions of this question are:

• What if we ask for $f_1$ to be a large irrational rotation -- can we still avoid introducing any new fixed points?
• What can we avoid introducing periodic points as well? How does this interact with the rotation number of $f_1$?
• If we cannot avoid introducing fixed points or periodic points, can we at least arrange for them to be hyperbolic? (The simplest examples of $f$ as required in this question naturally have extra elliptic fixed points on the interior of the annulus.) For this variant on the question we should ask for the extra fixed points to be nondegenerate.

I do not think one can answer any of the above questions affirmatively via the time-1 flow of a time independent hamiltonian, although I would be happy to be corrected on this point.

Answer 1

Can one find an $f$ with $f_1$ a small irrational rotation such that $f$ has no fixed points except for the fixed points of $f_0$, which we assume is a nondegenerate circle diffeomorphism of rotation number zero?

No, if $f_0$ has $k$ fixed points then $f$ must have at least $k/2$ extra fixed points, because of a fixed point index argument: Assume there are no fixed points other than the ones of $f_0$, which are nondegenerate. Doing a reflection (gluing a symmetric copy of the annulus) you can extend $f$ to a symplectomorphism of $[-1,1]\times \{0\}$ whose only fixed points are those in $S^1\times \{0\}$, which are hyperbolic saddles. The fixed point index of for each one of those is $-1$, while the total sum of the indices of $f$ must be $0$ due to the Leftschetz theorem. Also, the index of any fixed point of $f$ (in the interior of the annulus) is at most $1$ due to the fact that $f$ preserves area, and the conclusion follows.

What can we avoid introducing periodic points as well? How does this interact with the rotation number of $f_1$

Every rational number between the rotation numbers of $f_0$ and $f_1$ corresponds to a periodic point (note that if $p/q$ is such a number, applying the generalization of Poincaré-Birkhoff for fixed points to the lift $F^q - (p,0)$ of $f^q$ guarantees the existence of such periodic points). A good reference for this this article by Franks).

If we cannot avoid introducing fixed points or periodic points, can we at least arrange for them to be hyperbolic?

You can arrange for the fixed points to be hyperbolic. Below is an outline of an example.

The trick is that the fixed point $p$ has index $1$, i.e. it has negative eigenvalues, so it permutes the stable/unstable branches. So $A$ and $B$ form a period-two orbit). I think you could make all periodic points hyperbolic, but that would be much more delicate and unfortunately I don't know a reference.