A Full Study of the Dynamics on One-Holed Dilation Tori

In line with [1], we define dilation surfaces in the following way:

We can always find a polygonal model for the dilation surfaces of interest: a collection of polygons with sides identified in parallel opposite pairs by translations and/or dilations by real factors. For our purposes, we allow dilation surfaces to have boundary components which have no identifications. Some examples are found in Figures 2.1 and 2.2.

We can then consider the set of directions $\mathbb{RP}^{1}$ on the surface. For each direction $m\in\mathbb{RP}^{1}$, the foliation on $X$ by lines of slope $m$ is called the directional foliation of $X$ in direction $m$. The leaves of the directional foliation are the orbits of points under the straight-line or geodesic flow in direction $m$. Leaves which intersect a cone point terminate at that point. We call these leaves critical. A leaf that intersects cone points on both ends is called a saddle connection.

We are interested in the long-term dynamics of the geodesic flow on $X$. For a given directional foliation, we want to describe the $\omega$-limit (forward limit) and $\alpha$-limit (backward limit) sets of each noncritical leaf. For general dilation surfaces, this is a difficult task. In [5], it is conjectured that generically the foliations are Morse-Smale, i.e. attracted to closed leaves:

Note the assumption that $X$ not be a translation surface. In the case that $X$ is a translation surface, i.e. all transition maps of the atlas take the form $z\mapsto z+b$, then the generic behavior is determined in [6]:

In particular, such leaves are minimal and equidistributed. This is in stark contrast to the generic Morse-Smale behavior conjectured for dilation surfaces above. While a full proof of the conjectured generic dynamical behavior still seems far-off, it is possible to ascertain generic Morse-Smale behavior for simple classes of dilation surfaces. To do so, we pass to the machinery of affine interval exchange transformations.

To study dynamics of the straight line flow on a dilation surface, it sometimes helps to consider instead the dynamics of the first return map on a transversal to the flow. As we will see, this first return map will be an example of an affine interval exchange transformation. In this way, we can pass from studying dynamics of a flow on a surface to studying the dynamics of a map on the interval.

This technique of passing to a lower-dimensional dynamical system has also been used extensively in the study of translation surfaces. In the case of translation surfaces, the first return map to a transversal is an interval exchange transformation. A good introduction to this subject is [9].

Because there is some ambiguity as to where the endpoints of each interval map are mapped, we generally ignore the endpoints when thinking about AIETs.

It is informative to provide some examples of AIETs. We will use a graphical representation with the domain pictured on the top and the range on the bottom. $T$ carries each subinterval on top to the corresponding subinterval on bottom by an affine, orientation-preserving mapping. In Figure 2.3 we see an example of a surjective AIET where the original interval is split up into three pieces that are then permuted and scaled.

We can also consider AIETs that are not surjective. Figure 2.4 shows an AIET on three intervals where the image intervals do not cover the whole interval again.

To understand dynamics on a dilation surface in a particular direction, it is often sufficient to understand the dynamics of the first return map on a transversal to the direction of flow. In the case when the first return map exists for every point on the transversal, the first return map will always be an AIET. This correspondence is best seen through an example.

Suppose that we wanted to understand the dynamics of the vertical flow on the dilation surface in Figure 2.5. We might instead consider the first return map on the horizontal transversal shown. That is, for every point $x$ on the transversal, we let $T(x)$ be the point where the vertical flow from $x$ on the surface first returns back to the transversal.

If we repeat this process for every point on the transversal, we find that our first return map $T$ is the AIET on two intervals in Figure 2.5.

The dynamics of this $2$-AIET directly correspond to the dynamics of the vertical flow on the dilation surface since the vertical flow is a suspension of the $2$-AIET. If the $2$-AIET is minimal, has an attracting periodic orbit or critical orbit (orbit that eventually hits the break point $x_{T}$), is completely periodic, or has some other behavior, then the vertical flow on the dilation surface must have the same behavior. In this way, we have reduced our problem of understanding dynamics on a dilation surface to understanding the dynamics of a corresponding AIET.

First, we determine the behaviors of the flow in the boundary-parallel direction $m_{B}$. This flow is special in that it produces a bijective first return map on a transverse diagonal. We note that it may also be reversed without risk of hitting the boundary. The picture of the first return map $T$ of the flow on a diagonal is shown in Figure 3.4.

We find that the first return map is a bijective 2-AIET. In [4], a formula is provided to compute the rotation number of such a map. From the rotation number one can determine the dynamics. If the rotation number is irrational, then the dynamics are minimal. If the rotation number is rational, the dynamics may be either completely periodic or Morse-Smale. Once the dynamical behavior of $T$ is understood, the directional flow on $X$ is a suspension of this map and shares the same behavior.

The first return maps on some transverse diagonal to directions in $I_{1}$ are given by non-surjective 2-AIETs of a specific form. These are maps which swap the two intervals, expand one and contract the other, and place them against the left and right endpoints of the codomain. At first this desired form is not immediately apparent, but we may attain this form after applying the first return map once and restricting to the image. See Figure 3.5 for an example of this discussion.

The dynamics of maps which look like this will be studied in Section 4.4. We note here that maps in $I_{5}$ are of the same form as those in $I_{1}$, except for the choice of which interval expands and which contracts. By reflecting, we will see that the dynamics of maps in $I_{5}$ will be the mirror image of the dynamics of maps in $I_{1}$.

Trajectories moving in directions from $I_{2}$ are caught within an immediately present subsurface called a dilation cylinder, an affine cylinder with holonomy across its core curve.

There are two obvious dilation cylinders present in the polygonal model for $X$. The subsurface which consists of the convex hull of the two edges labeled $B$ in the polygonal model is the dilation cylinder corresponding to $I_{2}$.

This dilation cylinder is seen in the first return map. In Figure 3.6, it is seen that flows in $I_{2}$ after one iteration only land on the red interval $B$. The first return map is an injective affine contraction, so it has a unique fixed point.

This tells that the forward orbit from any noncritical point moving in a direction in $I_{2}$ is attracted to a closed orbit in the dilation cylinder. Note also that at each boundary direction of $I_{2}$, the directional foliation is attracted to a saddle connection.

Directions in $I_{4}$ obey similar behavior, captured in the dilation cylinder which is the convex hull of the two edges labeled $A$. We conclude that every directional foliation in the interiors of $I_{2}$ and $I_{4}$ converges to a closed orbit.

The directions in $I_{3}$ have a similar structure to those in $I_{1}$ and $I_{5}$, in the sense that their first return maps are injective 2-AIETs. They swap the two intervals, contracting both, and place them against the left and right endpoints of the codomain. These maps will immediately have the desired structure without needing to restrict to the image. See Figure 3.7.

We call maps like the first return maps in the intervals $I_{1}$, $I_{3}$, and $I_{5}$ $(\rho_{A},\rho_{B})$-maps. These maps will be defined (and their name justified) in the next section. While they are similar in structure to their surjective cousins, found in the boundary-parallel direction, their non-surjectivity leads to wildly different dynamics. To understand the dynamics of flows on dilation tori, it is essential that we analyze the dynamics of these maps in detail.

Our main tool in analyzing the dynamics of $(\rho_{A},\rho_{B})$-maps will be a form of Rauzy-Veech induction. Rauzy-Veech induction (often shortened as Rauzy induction) was originally defined for interval exchange transformations (IETs) and is a way of accelerating the dynamics of these maps. This way, we can keep track of the trajectory of a point in the original IET if we understand its trajectory in the accelerated map. For an introduction to Rauzy induction on IETs, see [9].

The Rauzy induction algorithm has also been used by various authors (e.g. [7]) to study surjective AIETs, and in [2], Boulanger and Ghazouani define a version of Rauzy induction for nonsurjective 2-AIETs that we will briefly summarize here.

Let $T:D\to D$ be a $(\rho_{A},\rho_{B})$-map. We will then consider the first return map of the AIET to a carefully chosen subinterval via a Rauzy induction step. One step of Rauzy induction produces another $(\rho_{A}^{\prime},\rho_{B}^{\prime})$-map where $\rho_{A}^{\prime}$ and $\rho_{B}^{\prime}$ are products of the original $\rho_{A}$ and $\rho_{B}$. Multiple steps of Rauzy induction starting with a map in $\mathcal{I}(\rho_{A},\rho_{B})$ then give us a collection of maps in the spaces $\mathcal{I}(\rho_{A}^{m_{A}}\rho_{B}^{m_{B}},\rho_{A}^{n_{A}}\rho_{B}^{n_{B}})$ for integers $m_{A},m_{B},n_{A},n_{B}\geq 0$. This leads us to the following inductive definition of Rauzy induction:

Suppose $T\in\mathcal{I}(\rho_{A}^{m_{A}}\rho_{B}^{m_{B}},\rho_{A}^{n_{A}}\rho_{B}^{n_{B}})$, let $A$ and $B$ be the left and right top intervals of $T$ respectively, and let $\lambda_{A},\lambda_{B}$ be their lengths. There are three possibilities:

1. (1)
   1. (R)

      (Right Rauzy Induction) $B\subsetneq T(A)$, i.e. $\lambda_{B}<\rho_{A}^{m_{A}}\rho_{B}^{m_{B}}\lambda_{A}$
      Letting $D^{\prime}=D\setminus B$, we consider the first return map $T^{\prime}$ on $D^{\prime}$. Then:
      $T^{\prime}\in\mathcal{I}(\rho_{A}^{m_{A}}\rho_{B}^{m_{B}},\rho_{A}^{m_{A}+n_{A}}\rho_{B}^{m_{B}+n_{B}})$ (recall that we may rescale the domain)
      and the lengths become $\lambda^{\prime}_{A}:=\lambda_{A}-\rho_{A}^{-m_{A}}\rho_{B}^{-m_{B}}\lambda_{B}$ and $\lambda^{\prime}_{B}:=\rho_{A}^{-m_{A}}\rho_{B}^{-m_{B}}\lambda_{B}$.
      This corresponds to the transformation $\lambda^{\prime}=R_{m_{A},m_{B}}\lambda$ where

      $$\lambda:=\begin{pmatrix}\lambda_{A}\\ \lambda_{B}\end{pmatrix},\qquad\lambda^{\prime}:=\begin{pmatrix}\lambda^{\prime}_{A}\\ \lambda^{\prime}_{B}\end{pmatrix},\qquad R_{m_{A},m_{B}}:=\begin{pmatrix}1&-\rho_{A}^{-m_{A}}\rho_{B}^{-m_{B}}\\ 0&\rho_{A}^{-m_{A}}\rho_{B}^{-m_{B}}\end{pmatrix}$$

      An example of right Rauzy induction is shown in Figure 4.2.

   2. (b)

      (Left Rauzy Induction) $A\subsetneq T(B)$, i.e. $\lambda_{A}<\rho_{A}^{n_{A}}\rho_{B}^{n_{B}}\lambda_{B}$

      Letting $D^{\prime}=D\setminus A$, we consider the first return map $T^{\prime}$ on $D^{\prime}$. Then:
      $T^{\prime}\in\mathcal{I}(\rho_{A}^{m_{A}+n_{A}}\rho_{B}^{m_{B}+n_{B}},\rho_{A}^{n_{A}}\rho_{B}^{n_{B}})$ and the new lengths are $\lambda^{\prime}=L_{n_{A},n_{B}}\lambda$ where

      $$\lambda:=\begin{pmatrix}\lambda_{A}\\ \lambda_{B}\end{pmatrix},\qquad\lambda^{\prime}:=\begin{pmatrix}\lambda^{\prime}_{A}\\ \lambda^{\prime}_{B}\end{pmatrix},\qquad L_{n_{A},n_{B}}:=\begin{pmatrix}\rho_{A}^{-n_{A}}\rho_{B}^{-n_{B}}&0\\ -\rho_{A}^{-n_{A}}\rho_{B}^{-n_{B}}&1\end{pmatrix}$$

      An example of a left Rauzy induction step is shown in Figure 4.3.

2. (2)

   (Termination) $T(A)\subseteq B$ and $T(B)\subseteq A$, i.e. $\lambda_{A}\geq\rho_{A}^{n_{A}}\rho_{B}^{n_{B}}\lambda_{B}$ and $\lambda_{B}\geq\rho_{A}^{m_{A}}\rho_{B}^{m_{B}}\lambda_{A}$

   Letting $D^{\prime}=D\setminus T(A)$, we consider the first return map $T^{\prime}$ on $D^{\prime}$. The resulting map has image in $A$, and is a contraction in $A$, so orbits under the map converge to an attractive fixed point in $A$ of derivative $\rho_{A}^{m_{A}+n_{A}}\rho_{B}^{m_{B}+n_{B}}$. The picture is in Figure 4.4.

Our goal now is to understand when the process of iteratively applying Rauzy induction to our original $(\rho_{A},\rho_{B})$-map terminates after finitely many steps, since terminating maps have an attracting periodic orbit or critical orbit (orbit that eventually hits the break point $x_{T}$).

We begin with an empty word $w$ and a map $T\in\mathcal{I}(\rho_{A},\rho_{B})$. If a Rauzy induction step is possible, we apply the appropriate (R) or (L) induction and record the letter $R$ or $L$ at the end of $w$. We repeat on the resulting maps either infinitely or until we reach a termination step.

We assume that our initial $(\rho_{A},\rho_{B})$ map $T$ is a map of the unit interval $[0,1]$ to itself. We then parameterize $(\rho_{A},\rho_{B})$-maps by the position of their breakpoint $x_{T}$ in $[0,1]$. For a finite word $w$, we let $I(w)\subset[0,1]$ be the set of maps that follow the word $w$ in their sequence of left and right induction steps. We also let $H(w)\subset I(w)$ be the set of maps that follow word $w$ and whose next step is a termination step.

The outline of our argument will go as follows:

1. 1.

   We will first consider the contracting case when $\rho_{A},\rho_{B}<1$. In this case, we will explicitly find the intervals of discontinuity points $H(w)\subset I(w)\subset[0,1]$ parameterizing those maps in $\mathcal{I}(\rho_{A},\rho_{B})$ where the Rauzy induction steps follow the sequence of inductions given by the word $w$, or terminate after that sequence.

2. 2.

   Building off of our work in the previous step, we will find that $|H(w)|/|I(w)|$ is uniformly bounded below. Since the set of parameters $H(w)$ where the Rauzy induction terminates is constructed like the complement of the middle thirds Cantor set, this bound will show that the terminating parameters form a full measure subset of the interval whose complement is a Cantor set.

3. 3.

   Finally, we will move on to the case when one of $\rho_{A}$ or $\rho_{B}$ is greater than $1$. We will find that with the exception of one starting parameter, the Rauzy induction will eventually land us in a terminating case or in a case where our scaling factors $\rho^{\prime}_{A}$ and $\rho^{\prime}_{B}$ are both contracting. We can then use our work from the $\rho_{A},\rho_{B}<1$ case to again conclude that the terminating parameters form a full measure subset of the interval that is the complement of a Cantor set.

4. 4.

   Using the results in steps 2 and 3, we can lift the attracting orbits on these $(\rho_{A},\rho_{B})$-maps to attracting orbits on corresponding one-holed dilation tori to provide a proof of Theorem 1.2.

We start with the case where $\rho_{A},\rho_{B}<1$. Let $T\in\mathcal{I}(\rho_{A},\rho_{B})$ where we have lengths $\lambda_{A}=x_{T}$, $\lambda_{B}=1-x_{T}$ for $x_{T}\in(0,1)$. We can consider a word $w$ of length $\ell$, $w=w_{1}\cdots w_{\ell}$ with $w_{i}\in\{R,L\}$, encoding a sequence of Rauzy induction moves. We associate to $w$ sequences of integers $m_{A}(i),m_{B}(i),n_{A}(i),n_{B}(i)$ with $(m_{A}(0),m_{B}(0),n_{A}(0),n_{B}(0))=(1,0,0,1)$ and the recurrence

Let $(m_{A},m_{B},n_{A},n_{B}):=(m_{A}(\ell),m_{B}(\ell),n_{A}(\ell),n_{B}(\ell))$ be the final term. Then upon applying the Rauzy inductions in $w$ to $T$, the resulting map is in $\mathcal{I}(\rho_{A}^{m_{A}}\rho_{B}^{m_{B}},\rho_{A}^{n_{A}}\rho_{B}^{n_{B}})$. We can define a sequence of length transformation matrices $M_{i}$ with $M_{0}=I$ and

where the $R$ and $L$ matrices are defined as in Section 4.1, so that the lengths of the final intervals up to scaling are $\begin{pmatrix}\lambda_{A}^{\prime}\\ \lambda_{B}^{\prime}\end{pmatrix}:=M_{\ell}\begin{pmatrix}\lambda_{A}\\ \lambda_{B}\end{pmatrix}=M_{\ell}\begin{pmatrix}x_{T}\\ 1-x_{T}\end{pmatrix}$.

Note that the sequence of moves $w$ is the valid sequence of Rauzy inductions executed on $T$ if and only if $\lambda^{\prime}_{A},\lambda^{\prime}_{B}\geq 0$. Indeed, if the sequence is invalid then the result of some step will give a negative entry in the length vector. Once one entry of the length vector is negative, it is easy to see that further multiplications by the $L$ and $R$ matrices keep some entry negative.

Suppose the final matrix is $M_{\ell}=:\begin{pmatrix}a&b\\ c&d\end{pmatrix}$. The following lemma will help us identify the parameters corresponding to a particular set of Rauzy induction moves.

This lemma allows us to identify the $(\rho_{A},\rho_{B})$-maps which undergo a sequence of Rauzy inductions corresponding to the word $w$:

Within $I(w)$, there is a subset $H(w)$ consisting of discontinuity points $x_{T}$ such that Rauzy induction on the corresponding map terminates after the sequence of moves $w$. In fact, $H(w)$ is a subinterval of $I(w)$ which may be determined in terms of $M_{\ell}$:

Note that when $w\neq v$, $H(w)\cap H(v)=\emptyset$. This is because the Rauzy induction algorithm has no ambiguity, there is only one valid induction word of length $k$ for any given $(\rho_{A},\rho_{B})$-map. Let $H_{k}:=\bigcup_{|w|\leq k}H(w)$ parameterize the set of all $(\rho_{A},\rho_{B})$-maps which terminate after at most $k$ Rauzy inductions, and $H=\bigcup_{k}H_{k}$ parameterize the set of all $(\rho_{A},\rho_{B})$-maps which terminate at all under the Rauzy algorithm. $H$ has the same construction as the complement of the Cantor triadic set. Indeed, $H_{k}$ adds $2^{k}$ intervals, one within each gap between the intervals in $H_{k-1}$. These gaps are the intervals $I(w)$ where $|w|=k$. For some $(\rho_{A},\rho_{B})$-map, $H_{2}$ might look like the set in Figure 4.5.

For every $x_{T}\in H$, since the Rauzy induction terminates after finitely many steps, we can find a periodic orbit for $T$ beginning at the attractive fixed point of the final map $T^{\prime}$. This orbit then attracts all other orbits and corresponds to an attracting orbit for the directional foliation on a corresponding dilation surface.

We now check that $H$ is of full measure in $[0,1]$. This is where we utilize $\rho_{A},\rho_{B}<1$. The following intermediate bound does much of the heavy lifting:

With the help of this bound, we may bound $|H(w)|/|I(w)|$ below:

Now, we can bring these results back to the dilation torus $X$ to prove part of Theorem 1.2. Recall that on $X$ we have yet to understand the dynamics of directional foliations in three intervals of directions introduced in Section 3: $I_{1}$, $I_{3}$, and $I_{5}$. What we have proven thus far handles the dynamics in $I_{3}$.