Centralizers in the R. Thompson group $V_{n}$

If $\alpha\in V_{n}$, we set

We call the group $C_{V_{n}}(\alpha)$ the centralizer in $V_{n}$ of $\alpha$, as is standard.

Our primary theorem is the following.

We now explain the statement of this theorem in a bit more detail.

The group $\langle\alpha\rangle$ acts on a subset of the nodes of the infinite $n$-ary tree. The number $s$ represents the number of distinct lengths of finite cyclic orbits of nodes under this action. The value of $s$ is easy to compute from any given revealing pair representing $\alpha$.

For the action mentioned above, each $r_{i}$ is determined as a minimal number of nodes carrying a fundamental domain (for the action of a conjugate version of $\langle\alpha\rangle$) in the set of nodes supporting the cycles of length $m_{i}$.

For each $m_{i}$, we have $K_{m_{i}}=(Maps(\mathfrak{C}_{n},\mathbb{Z}_{m_{i}}))^{r_{i}}$, where $Maps(\mathfrak{C}_{n},\mathbb{Z}_{m_{i}})$ is the group of continuous maps from $\mathfrak{C}_{n}$ to $\mathbb{Z}_{m_{i}}$ under point-wise multiplication, and where $\mathbb{Z}_{m_{i}}$ is the cyclic group $\mathbb{Z}/(m_{i}\mathbb{Z})$ under the discrete topology. We note that $K_{m_{i}}$ is not finitely generated for $m_{i}>1$.

The groups $G_{n,r_{i}}$ are the Higman-Thompson groups from [14].

Given any element $\beta\in V_{n}$, one can associate an infinite collection of finite labeled graphs (which we call flow graphs). Flow graphs are labeled, directed, finite graphs and which describe structural meta-data pertaining to the dynamics of certain subsets of $\mathfrak{C}_{n}$ under the action of $\langle\alpha\rangle$. Flow graphs are themselves “quotient objects” coming from discrete train tracks, which are objects we introduce here to model dynamics in the Cantor set much as regular train tracks model dynamics in surface homeomorphism theory [22, 23, 18]

Components of a flow graph associated with $\alpha$ fall into equivalence classes $\{ICC_{i}\}$ which model similar dynamics. The number $t$ is the number of equivalence classes of components carrying infinite orbits under the action of $\langle\alpha\rangle$. This number happens to be independent of the representative flow graph chosen.

The right factor of the main direct product represents the restriction of the centralizer of $\alpha$ to elements which are the identity away from the closure of the region where $\langle\alpha\rangle$ acts with non-finite orbits.

For each $j$, the supports of the elements of $ICC_{j}$ represent regions in the action of $\langle\alpha\rangle$ where the centralizers of $\alpha$, with restricted actions to these regions, are isomorphic. For any such support of an element of $ICC_{j}$, the finite order elements of the restricted centralizer form a group. We take $A_{j}$ to be a representative group from set of these isomorphic finite groups.

The group $P_{q_{j}}$ is the full symmetric group on the $q_{j}$ isomorphic flow graph components in $ICC_{j}$.

Recall now that by Higman in [14], the groups $G_{k,r}$ are all finitely presented. This fact, together with a short analysis of the nature of the actions in the left-hand semi-direct products, shows that each of the groups $K_{m_{i}}\rtimes G_{n,r_{i}}$ are finitely generated (see Corollary 6.2 below). As the groups on the right-hand side of the central direct product are manifestly finitely generated, we obtain the following corollary to Theorem 1.1.

One could try to improve this last corollary to obtain a statement of finite-presentation, which may be feasible. Such a proof might be accomplished through a careful study of presentations of the Higman groups $G_{n,r}$. In this direction, we ask the following.

We have also found some evidence supporting the possibility that the answer to the following question is “Yes.”

It is not completely trivial to find an example element $\alpha\in V=V_{2}$ where any $A_{i}$ is not cyclic. In section 7 we give such an example where $t=1$ and $A_{1}\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$.

A generator of each $\mathbb{Z}$ in the right hand terms $A_{i}\rtimes\mathbb{Z}$ is given by a root of a restricted version of $\alpha$ which is restricted to act only on the support of an element in $ICC_{j}$. We know that $A_{i}$ commutes with the restricted version of $\alpha$ by definition, but it is not clear that $A_{i}$ will commute with any valid choice of a generator for the $\mathbb{Z}$ term.

In the final section of the paper we prove the following theorem.

We give a description of our approach to centralizers in the broadest terms.

Let $\alpha\in V_{n}$. Let $H=\langle\alpha\rangle$. Define the fundamental domain of the action of $H$ to be the space $\mathfrak{C}_{n}/H$. Note that this space is generally not very friendly, i.e., it is typically not Hausdorff, but that fact will have almost no bearing on our work.

If an element commutes with $\alpha$, it will induce an action on $\mathfrak{C}_{n}/H$.

Thus, we get a short exact sequence:

Here, $\mathcal{K}$ represent the elements in $C_{V_{n}}(\alpha)$ which act on $\mathfrak{C}_{n}$ in such a way that their induced action on $\mathfrak{C}_{n}/H$ is trivial. The group $\mathcal{Q}$ is the natural quotient of $C_{V_{n}}(\alpha)$ by the image of the inclusion map $\mathcal{K}\to C_{V_{n}}(\alpha)$. Loosely speaking, elements of $\mathcal{Q}$ are represented by elements of $V_{n}$ which act in the “same” way on each “copy” of the fundamental domain in $\mathfrak{C}_{n}$ (this of course is imprecise; there may be no embedding of a fundamental domain in $\mathfrak{C}_{n}$).

The first author is indebted to Martin Kassabov for pointing out this general structural approach to analyzing centralizers in groups of homeomorphisms.

The only material in this section that may be unfamiliar to readers conversant with R. Thompson group literature is some of the language describing the Cantor set underlying a node of the tree $\mathscr{T}_{n}$ and related concepts.

Our primary perspective will be to consider $V_{n}$ as a group of homeomorphisms of the Cantor set $\mathfrak{C}$. In particular, $V_{n}$ should be thought to act as a group of homeomorphisms of the Cantor set $\mathfrak{C}_{n}\cong\mathfrak{C}$. That is, the version $\mathfrak{C}_{n}$ of the Cantor set that is realized as the boundary of the standard infinite, rooted $n$-ary tree $\mathscr{T}_{n}$. While we assume the reader understands that realization of the Cantor set, and also the terms “node” or “vertex”, “child”, “parent”, “ancestor”, “descendant”, “leaf”, and “$n$-caret” (or simply “caret”), and similar related language when referring to aspects of rooted $n$-ary trees, we give a short discussion below to establish some of our standard usage.

The left-to-right ordering of the children of a vertex in $\mathscr{T}_{n}$ allows us to give a name to each vertex in $\mathscr{T}_{n}$. Given a vertex $p$ in $\mathscr{T}_{n}$, there is a unique order preserving bijection $\textrm{Ord}_{p}:Children(p)\to\left\{0,1,2,\ldots,n-1\right\}$. Let $v$ be a vertex of $\mathscr{T}_{n}$ and let $(v_{i})_{i=1}^{m}$ be the unique descending path in $\mathscr{T}_{n}$ starting at the root $r=v_{1}$ and ending at $v=v_{m}$, then we name the vertex $v$ with the sequence $(\textrm{Ord}_{v_{i}}(v_{i+1}))_{i=1}^{m-1}$. We will think of this sequence as a string (ordered from left-to-right).

Given names of two vertices, we may concatenate these strings to produce the name of a third vertex which will be a descendant of the first vertex

Below, we diagram an example of $\mathscr{T}_{2}$, with a finite tree $T$ highlighted within it. The vertex $c$ is a leaf of $T$, and in both $T$ and $\mathscr{T}_{2}$, $c$ is a child of $b$ which is a child of $a$. The vertex $a$ is an ancestor of $c$ and $c$ is a descendant of $a$. The name of the vertex labeled by $c$ is $010$.

We can view a finite rooted $n$-ary tree $T$ as an instruction on how to partition the Cantor set $\mathfrak{C}_{n}$. Consider the natural embedding of $T$ into the tree $\mathscr{T}_{n}$, where we send the root of $T$ to the root of $\mathscr{T}_{n}$, and we preserve orders of children. For instance, as in the diagram above. Now, consider the set $P_{n}$ of all infinite descending paths in $\mathscr{T}_{n}$ which start at the root of $\mathscr{T}_{n}$. If we consider each ordered $n$-caret in $\mathscr{T}_{n}$ as an instruction to pass through another inductive subdivision of the unit interval in the formation process of $\mathfrak{C}_{n}$, then each element in $P_{n}$ can be thought of as limiting on an element of $\mathfrak{C}_{n}$. We thus identify $P_{n}$ with $\mathfrak{C}_{n}$. The set $P_{n}$ will now be considered to be topologized using the induced topology from the metric space topology of the unit interval. Now if we consider a vertex $c$ of $T$, we can associate $c$ with the subset of $\mathfrak{C}_{n}$ corresponding to the paths in $P_{n}$ which pass through $c$. We will call this the Cantor set under $c$, and we will call any such subset of $\mathfrak{C}_{n}$ an interval of $\mathfrak{C}_{n}$. It is immediate that any interval in $\mathfrak{C}_{n}$ is actually homeomorphic with $\mathfrak{C}_{n}$. Given a node $c$, the Cantor set under $c$ is also commonly called a cone neighborhood in $\mathfrak{C}_{n}$, and by definitions these sets form the standard basis for the product topology on $P_{n}=\{0,1,\ldots,n-1\}^{\omega}\cong\mathfrak{C}_{n}$. Thus, the leaves of $T$ partition the set $P_{n}$, and they also partition $\mathfrak{C}_{n}$, into a set of open basis sets. We will call this the partition of $\mathfrak{C}_{n}$ associated with the tree $T$.

Extending the language of the previous paragraph, given a set $X\subset\mathfrak{C}_{n}$, we will call any node $c$ of the universal tree $\mathscr{T}_{n}$ which has its underlying set contained in $X$, a node of $X$.

Of course, for all integers $m,n>1$, we have that $\mathfrak{C}_{m}\cong\mathfrak{C}_{n}\cong\mathfrak{C}_{2}=\mathfrak{C}$.

Using the example above, the interval of $\mathfrak{C}_{2}$ under $c$ is $\mathfrak{C}_{2}\cap[2/9,7/27]$. In discussion, we will generally not distinguish between a vertex of $\mathscr{T}_{n}$ and the interval under it.

Some of the language in this subsection is unusual, although the general content will be familiar to all readers with knowledge of the R. Thompson groups.

An element of $\textrm{Homeo}(\mathfrak{C}_{n})$ is allowable if it can be represented by an allowable triple $(A,B,\sigma)$. The triple $(A,B,\sigma)$ is allowable if there is a positive integer $m$ so that $A$ and $B$ are rooted, finite, $n$-ary trees with the same number $m$ of leaves, and $\sigma\in\Sigma_{m}$, the permutation group on the set $\left\{1,2,3,\ldots,m\right\}$. We explain below how to build the homeomorphism $\alpha$ which is represented by such an allowable triple $(A,B,\sigma)$. We then call $(A,B,\sigma)$ a representative tree pair for $\alpha$. The group $V_{n}$ consists of the set of all allowable homeomorphisms of $\mathfrak{C}_{n}$ under the operation of composition.

We are now ready to explain how an allowable triple $(A,B,\sigma)$ defines an allowable homeomorphism $\alpha$ of $\mathfrak{C}_{n}$. Suppose $A$ and $B$ both have $m$ leaves, for some integer $m>1$. We consider $A$ to represent the domain of $\alpha$, and $B$ to represent the range of $\alpha$. We take the leaves of $A$ and $B$ and number them in their natural left-to-right ordering from $1$ to $m$. For each index $i$ with $1\leq i\leq m$, we map the interval under leaf $i$ of $A$ to the interval under the leaf $i\sigma$ of $B$ using an orientation-preserving affine homeomorphism (that is, the homeomorphism of the two Cantor sets underlying these leaves defined by a restriction and co-restriction of an affine map with positive slope from the real numbers $\mathbb{R}$ to $\mathbb{R}$). Note for $\mathfrak{C}_{n}$ the slopes of such maps will be integral powers of $2n-1$. We will say that the leaf $i$ of $A$ is mapped to the leaf $i\sigma$ of $B$ by $\alpha$.

The next remark follows immediately from the discussion above:

The diagram below illustrates an example for $V_{2}=V$. The tree $A$ is on the left, and the tree $B$ is on the right. Note that we have re-decorated the leaves of $B$ with the numbering from $\sigma$. The diagram beneath the tree pair is intended to indicate where intervals of $\mathfrak{C}_{2}$ are getting mapped, where the intervals $D$ represent the domain and the intervals $R$ represent the range. The lower diagram is superfluous when defining a general element of $V$.

In general, as we can decorate our tree leaves with numbers to indicate the bijection, we will now re-define the phrase “tree pair” throughout the remainder of the paper to mean an allowable triple. We will still discuss the permutation of a tree pair as needed.

As mentioned in the introduction, there are groups $F_{n}\leq T_{n}\leq V_{n}$. If we only allow cyclic permutations, we get the group $T_{n}$. If our permutation is trivial, we get $F_{n}$. Thus, we can think of $F_{n}$ as a group of piecewise-linear homeomorphisms of the interval $[0,1]$, while $T_{n}$ can be thought of as a group of piecewise-linear homeomorphisms of the circle $S^{1}$.

Here is an example element $\theta\in T=T_{2}$, which we will be considering again later.

Below is an example element from $F=F_{2}$.

There is nothing new in this subsection; readers familiar with the R. Thompson groups should skip ahead.

Naturally, the group operation of $V_{n}$ is given by composition of functions. Building compositions using tree pairs is not hard. The process is enabled by the fact that there are many representative tree pairs for an element of $V_{n}$.

We give an example of what we mean by the last sentence. Given a representative tree pair $(A,B,\sigma)$, one can use a leaf $i$ of $A$ to be a root of an extra $n$-caret, creating $A^{\prime}$, and one can build a tree $B^{\prime}$ from $B$ by replacing the leaf $i\sigma$ with an $n$-caret. Label the leaves of $A^{\prime}$ in increasing order, and the leaves of $B^{\prime}$ using the induced labeling from the permutation $\sigma$ on all leaves of $B^{\prime}$ that are also leaves of $B$. For the other leaves of $B^{\prime}$, use the labeling, in order, of the $n$-caret in $A^{\prime}$ that is not a $n$-caret of $A$. Let us call the permutation we have built from the leaves of $A^{\prime}$ to the leaves of $B^{\prime}$ by $\sigma^{\prime}$. The process of replacing the tree pair $(A,B,\sigma)$ by $(A^{\prime},B^{\prime},\sigma^{\prime})$ is called a simple augmentation.

If the reverse process can be carried out (that is, deleting an $n$-caret from both the domain and range trees and re-labeling the permutation so that our initial two trees appear as a simple augmentation of our resulting tree pair) then we call this process a simple reduction. If we carry out either a simple augmentation or a simple reduction, we may instead say we have done a simple modification to our initial tree pair.

The following lemma is a straightforward consequence of the standard fact that any element in $V_{n}$ has a unique tree pair representation that will not admit any simple reductions.

We are now ready to carry out multiplication of tree pairs. The essence of the idea is to augment the range tree of the first element and the domain tree of the second element until they are the same tree (and carry out the necessary augmentations throughout both tree pairs), and then re-label the permutations, using the labeling of the range tree of the first element to seed the re-labeling of the permutation in the second element’s tree pair. At this junction, the two inner trees are completely identical, and can be removed. The diagram below demonstrates this process.

Thus, we now know what $V_{n}$ is, and we can represent and multiply its elements.

Throughout this section, we will work with some nontrivial $\alpha\in V_{n}$. We will assume that the tree pair $(A,B,\sigma)$ represents $\alpha$.

Consider $A$ and $B$ as finite rooted $n$-ary sub-trees (with roots at the root of $\mathscr{T}_{n}$) of $\mathscr{T}_{n}$.

We will call the set of vertices of $\mathscr{T}_{n}$ which are leaves of both $A$ and $B$ the neutral leaves of $(A,B,\sigma)$. We will simply call these the neutral leaves, if the tree pair is understood.

As both $A$ and $B$ have a root and neither are empty, we can immediately form the tree $C=A\cap B$. It is immediate that the neutral leaves are leaves of $C$, but if $A\neq B$, then $C$ will have other leaves as well.

We can make the sets $X=A-B$ and $Y=B-A$. The closures $\overline{X}$ and $\overline{Y}$ in $\mathscr{T}_{n}$ are both finite disjoint unions rooted $n$-ary trees (their roots are not sitting at the root of $\mathscr{T}_{n}$), where the number of carets in $\overline{X}$ is the same as the number of carets of $\overline{Y}$ (this number could be zero if $A=B$). We call $\overline{X}$ and $\overline{Y}$ a difference of carets for $A$ and $B$.

In the remainder, when we write $D-E$, where $D$ and $E$ are trees, we actually want to take the difference of carets, so that our result will be a collection of rooted trees.

While $\langle\alpha\rangle$ acts on the Cantor set, it also induces a “partial action” on an infinite subset of the vertices of $\mathscr{T}_{n}$, as we explain in this paragraph. Since the interval under a leaf $\lambda$ of $A$ is taken affinely to the interval under a leaf of $B$ (we will denote this leaf by $\lambda\alpha$), we see that $\alpha$ induces a map from the vertices of $\mathscr{T}_{n}$ under $\lambda$ to the vertices of $\mathscr{T}_{n}$ under $\lambda\alpha$. In particular, the full sub-tree in $\mathscr{T}_{n}$ with root $\lambda$ is taken to the full sub-tree in $\mathscr{T}_{n}$ with root $\lambda\alpha$ in order preserving fashion. We note in passing that we cannot extend this to a true action on the vertices of $\mathscr{T}_{n}$; if we consider a vertex $\eta$ in $\mathscr{T}_{n}$ which is above a leaf of $A$, the map $\alpha$ may take the interval underlying $\eta$ and map it in a non-affine fashion across multiple intervals in $\mathfrak{C}_{n}$.

As an example of the behavior mentioned above, consider the element $\theta$ again. The parent vertex of the domain leaves labeled 3 and 4 is mapped across multiple vertices of the range tree, in a non-affine fashion.

If $\eta$ is a vertex of $\mathscr{T}_{n}$ we can now define a forward and backward orbit $O_{\eta}$ of $\eta$ in $\mathscr{T}_{n}$ under the action of $\langle\alpha\rangle$, to the extent that we restrict ourselves to powers of $\alpha$ that take the interval under $\eta$ affinely bijectively to an interval under a vertex of $\mathscr{T}_{n}$. In more general circumstances, given any integer $k$ and a vertex $\eta$ of $\mathscr{T}_{n}$, we will use the notation $\eta\alpha^{k}$ to denote the subset of $\mathfrak{C}_{n}$ which is the image of the interval under $\eta$ under the map $\alpha^{k}$, and if that set happens to be the interval under a vertex $\tau$ in $\mathscr{T}_{n}$ (so that the restriction of $\alpha^{k}$ to the interval under $\eta$ takes the interval underlying $\eta$ affinely to the interval underlying $\tau$), then we may denote the vertex $\tau$ by $\eta\alpha^{k}$ as well.

Let $L_{(A,B,\sigma)}$ denote the set of vertices of $\mathscr{T}_{n}$ which are either leaves of $A$ or leaves of $B$, and let $\lambda\in L_{(A,B,\sigma)}$.

It is possible that $\lambda$ is a neutral leaf whose vertex in $\mathscr{T}_{n}$ has orbit $O_{\lambda}$ entirely contained in the neutral leaves of $A$ and $B$. By Remark 2.2, and the fact that the neutral leaves are finite in number, we see that in this case the orbit of $\lambda$ is periodic in $\mathscr{T}_{n}$. In this case, we call $\lambda$ a periodic leaf.

Now suppose $\lambda$ is not a periodic leaf of $A$. This implies that if we consider the forward and backward orbits of $\lambda$ under the action of $\langle\alpha\rangle$, then in both directions, the orbit will exit the set of neutral leaves. In particular, there is a minimal integer $r\leq 0$ and a maximal integer $s\geq 0$ so that for all integers $i$ with $r\leq i\leq s$ we have that $\lambda\alpha^{i}\in L_{(A,B,\sigma)}$. It is also immediate that $\lambda\alpha^{r}$ is a leaf of $A-B$ and $\lambda\alpha^{s}$ is a leaf of $B-A$, while for all values of $i$ with $r<i<s$ we see that $\lambda\alpha^{i}$ is a neutral leaf.

Thus, we have the following lemma.

We now define and comment on some language from [5] and from [21]. Suppose we have the hypotheses of Lemma 4.1. The definition of iterated augmentation chain $IAC(\lambda)$ in Lemma 4.1 reflects the fact that we can augment the trees $A$ and $B$ at each vertex along an iterated augmentation chain, and end up with a new representative tree pair for $\alpha$ (the first vertex in an augmentation chain of type $(2)$ or $(3)$ can only be augmented in the domain tree $A$, while the last such vertex can only be augmented in the range tree $B$).

If $\lambda$ is of type $(2)$, with $IAC(\lambda)=(\lambda\alpha^{i})_{i=0}^{s}$ for some positive integer $s$, where $\lambda\alpha^{s}$ is an ancestor of $\lambda$, then we say that $\lambda$ is a repeller. If $\lambda$ is of type (3), with $IAC(\lambda)=(\lambda\alpha^{i})_{i=r}^{0}$ for some negative integer $r$, and where $\lambda\alpha^{r}$ is an ancestor of $\lambda$, then we say that $\lambda$ is an attractor.

We are now ready to define what it means for a tree pair to be a revealing pair. Suppose $\alpha\in V$ and the tree pair $(A,B,\sigma)$ represents $\alpha$. If every component of $A-B$ contains a repeller, and every component of $B-A$ contains an attractor, then we say that $(A,B,\sigma)$ is a revealing pair representing $\alpha$.

The discussion beginning section 10.7 in [5] proves that every element of $V_{n}$ admits a revealing pair. It is not hard to generate an algorithm which will transform any representative tree pair for an element of $V_{n}$ into a revealing pair.

Given $\alpha\in V_{n}$, we will denote by $R_{\alpha}$ the set of all revealing pairs for $\alpha$. We will use the symbol $\sim$ as a relation in the fashion $(A,B,\sigma)\sim\alpha$ denoting the fact that $(A,B,\sigma)\in R_{\alpha}$. In this case, we can further name leaves of $A-B$ and $B-A$. If $\lambda$ is a leaf of $A-B$ and $\lambda$ is not a repeller, then we say $\lambda$ is a source. If $\lambda$ is a leaf of $B-A$ and $\lambda$ is not an attractor, then we say $\lambda$ is a sink.

There are finitely many process types called “Rollings” introduced by Salazar in [21]. Rollings are methods by which one can carry out a finite collection of simple expansions to a revealing tree pair $(A,B,\sigma)$ to produce a new revealing pair $(A^{\prime},B^{\prime},\sigma^{\prime})$.

Below, we give the definitions and some discussion for rollings of type II. We give definitions for the other types of rollings in sub-section 4.2.

The tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})$ is a single rolling of type II from $(A,B,\sigma)$ if it is obtained from $(A,B,\sigma)$ by adding a copy of a component $U$ of $A-B$ to $A$ at the last leaf in the orbit of the repeller in $U$ and to $B$ at its image; or, by adding a copy of a component $W$ of $B-A$ to $A$ at the first leaf in the orbit of the attractor (the leaf of $A$ corresponding to the root node of $W$ in $B$) and to $B$ at its image.

The tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})$ is a rolling of type II from $(A,B,\sigma)$ if it is obtained from $(A,B,\sigma)$ by a finite collection of single rollings of type II applied to the initial tree pair $(A,B,\sigma)$ in some order.

We now state three lemmas about properties of revealing pairs. All of these properties are fairly straightforward to verify. In the cases of Lemma 4.2 and Lemma 4.3, the curious reader may also refer to the discussion in Sections 3.3 and 3.4 of [21] for alternative proofs.

Below, we slightly abuse the notion of a vertex name, by associating an infinite descending path with an infinite “name” string. This then represents a point in the Cantor set which is the boundary of the infinite tree.

Our first lemma discusses repellers, sources, and fixed points.

Proof:

The inverse $\alpha^{-s}$ of $\alpha^{s}$ maps the Cantor set $X_{s}$ under $\gamma_{s}$ bijectively to the Cantor set $X_{0}$ under $\gamma_{0}$, where $X_{0}\subset X_{s}$, in affine fashion. The only infinite descending path which is fixed by this map is the path terminating in $\Gamma^{\infty}$, thus $p_{\gamma_{0}}=\gamma_{0}\Gamma^{\infty}=\gamma_{s}\Gamma\Gamma^{\infty}$ is the unique attracting fixed point of $\alpha^{-s}$ in $X_{s}$, and is thus the unique repelling fixed point of $\alpha^{s}$ within $X_{0}$ (and even within all of $X_{s}$) under the action of $\alpha^{s}$.

We now show below why each of the points $p_{\gamma_{i}}$ are also repelling fixed points of $\alpha^{s}$.

We first obtain a new revealing tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})\sim\alpha$ which is a single rolling of type II from $(A,B,\sigma)$, constructed as follows.

Glue a copy of $C$ at $\lambda_{s}$ in $B$ to produce $B^{\prime}$, and a further copy of $C$ at the leaf $\gamma_{s-1}$ of $A$ to produce $A^{\prime}$. The resulting tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})$ so obtained has all of the leaves of $C$ of $A-B$ as neutral leaves (except in the case where $s=1$, in which case $A^{\prime}$ is $A$ with a copy of $C$ attached at $\gamma_{0}$, a leaf of the original $C$).

In any case, the original copy of $C$, within $A^{\prime}$, is now contained in $A^{\prime}\cap B^{\prime}$. The new copy of $C$ in $A^{\prime}$ is a complementary component of $A^{\prime}-B^{\prime}$ and contains the leaf $\gamma_{s-1}\Gamma$ as a repeller. There are no other new complementary components for the tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})$, which therefore must represent a revealing pair for $\alpha$.

Now, by a minor adjustment to the argument in first paragraph, the point $p_{\gamma_{s-1}}$ is a fixed repelling point of $\alpha^{s}$.

We can now continue inductively in this fashion to show that each point $p_{\gamma_{i}}$ is the unique repelling fixed point of $X_{i}$ under the action of $\alpha^{s}$ by building a revealing pair $(A^{\prime\prime},B^{\prime\prime},\sigma^{\prime\prime})$ for $\alpha$ with the point $p_{\gamma_{i}}$ as a point in a repelling leaf of a complementary component of $(A^{\prime\prime},B^{\prime\prime},\sigma^{\prime\prime})$ with shape $C$ and spine $\Gamma$ rooted at $\gamma_{i}$.

We leave the second point of the lemma to the reader, while the third point is immediate from the first since some power of $\alpha$ has a repelling fixed point.

$\diamond$

We call each subset $X_{i}$ a basin of repulsion for $\alpha$, since all points in $X_{i}$ eventually flow out of $X_{i}$ under repeated iteration of $\alpha^{s}$, never to return, except for $p_{\gamma_{i}}$, for each $0\leq i\leq s$. We call the string $\Gamma$ above the spine of the repeller $\gamma_{0}$ or the spine of $C$, as it describes the shape of the path in $C$ from the root of $C$ to the repeller $\gamma_{0}$.

We call each point $p_{\gamma_{i}}$ a periodic repelling point of $\alpha$ for each index $i$ with $0\leq i<s$. We may also call these points fixed repelling points of $\alpha^{s}$, noting in passing that not all periodic repelling points of $\alpha$ are necessarily fixed by $\alpha^{s}$. In similar fashion we call the sequence of points $(p_{\gamma_{i}})_{i=0}^{s-1}$ a periodic orbit of periodic repelling points for $\alpha$. We denote by $\mathcal{R}_{\alpha}$ the set of periodic repelling points of $\alpha$, noting that it is a finite set.

We now give a similar lemma discussing attractors and sinks.

Proof:

This proof is similar to the proof of the previous lemma, where here $\alpha^{-1}$ has a tree pair with $\lambda$ as a repeller, and each $p_{\gamma_{i}}$ is a periodic repelling point of $\alpha^{-1}$.

$\diamond$

We now extend the notation from Lemma 4.2 to apply to the sets named in Lemma 4.3 as below.

We call each subset $X_{i}$ a basin of attraction for $\alpha$ as indicated by $(A,B,\sigma)$, since all points in $X_{i}$ eventually limit to $p_{\gamma_{i}}$ under repeated iteration of $\alpha^{-r}$, for each $r\leq i\leq 0$. We call the string $\Gamma$ above the spine of the attractor $\gamma_{0}$ or the spine of $C$, as it describes the shape of the path in $C$ from the root of $C$ to the attractor $\gamma_{0}$.

We call each point $p_{\gamma_{i}}$ a periodic attracting point of $\alpha$ for each index $i$ with $r\leq i\leq 0$. We may also call these points fixed attracting points of $\alpha^{-r}$. In similar fashion we call the sequence of points $(p_{\gamma_{i}})_{i=r}^{-1}$ a periodic orbit of periodic attracting points for $\alpha$. We denote by $\mathcal{A}_{\alpha}$ the set of periodic attracting points of $\alpha$, noting that it is a finite set.

In the previous two lemmas, if $\lambda$ is a source or a sink (case two in each lemma), we refer to $IAC(\lambda)$ as a source-sink chain.

The next lemma follows directly from the two above, and the classification of the orbits of the leaves of $A$ and $B$. This lemma is a version of one result proved by Burillo, Cleary, Stein and Taback in their joint work [8] as Proposition 6.1. We give a new proof here, as the situation is greatly simplified through the use of revealing pairs.

Proof:

Suppose $\alpha$ does not have infinite order, and let $(A,B,\sigma)\sim\alpha$ be a revealing pair representing $\alpha$. We must have that $(A,B,\sigma)$ admits no repellers or attractors, thus both $A-B$ and $B-A$ are empty, and so $A=B$.

Suppose instead $(A,B,\sigma)$ is a tree pair representing $\alpha$ with $A=B$. Then it is immediate from the definition of multiplication for tree pairs that the order of $\alpha$ is the order of the permutation $\sigma$.   $\diamond$

We denote by $\mathcal{P}_{\alpha}$ the points of $\mathfrak{C}_{n}$ which underlie the periodic neutral leaves of $L_{(A,B,\sigma)}$. Note that the set $\mathcal{P}_{\alpha}$ is independent of the choice of revealing pair used to represent $\alpha$. We further denote by $Per(\alpha)$ the set of all periodic points of $\alpha$, that is

The following tree pairs represent the previously defined element $\theta\in T$ (we apply an augmentation to our first tree pair to produce a revealing pair representing $\theta$).

A quick examination of the second tree pair above should convince the reader that $\theta$ has order five and rotation number $2/5$.

We now give a series of diagrams for a revealing pair $(A,B,\sigma)$ representing a particular non-torsion element $\alpha\in V$. This element has a revealing pair which contains many of the structures we have been discussing. In each diagram below we illustrate some of the particular aspects we have discussed above.

Below is an example of our revealing pair $(A,B,\sigma)$, with the neutral leaves underlined. Recall that the left tree is $A$, and the right tree is $B$.

In the next diagram, we point out a periodic orbit of neutral leaves of length two.

Below, the vertex $11111$ in $B$ is an attractor; its iterated augmentation chain is $(111,11111)$, a sequence of length two. The fixed point of the attractor corresponds to the value $1$ in the unit interval. The word $\Gamma$ for this attractor is $11$.

In the diagram to follow, the vertices $0000$ and $011$ represent repellers in $A$. The dotted paths track the orbits of the repellers along their iterated augmentation chains.

Finally, we highlight the fact that sources flow to sinks. Note how the lengths of the paths from sources to sinks are not uniform. In particular, the source $0100$ first hops to $110$ before next landing in the basin of attraction under the vertex $000$.

In this subsection we are concerned with modelling the dynamics in $\mathfrak{C}_{n}$ under the action of the group generated by an element $v$ of $V_{n}$. Naturally, the relevant information is contained in any particular revealing pair $\rho=(C,D,\theta)$ for the element, but we have found that tools from surface topology afford us another way to visualize these dynamics. In particular, these dynamics can be usefully described by a combinatorial object introduced by Thurston (see [22, 23]) to aid in the study of surfaces, namely, a train track. Given a revealing pair, it is easy to draw a combinatorial train track which in turn “carries” a lamination in some compact surface with boundary. It is this lamination which models, in some sense, the movement of points in the Cantor set under iteration of the map $v$. As in the theory of laminations carried by a train track, one quickly realizes that most of the relevant visual information is actually contained in the train-track object (see for instance, Penner and Harer’s book [18]). We call the train track object developed here a discrete train track, even though it is continuous in nature; the name is meant to emphasize that our train tracks model dynamics in a totally disconnected set under iterations of a fixed map.

Below, we describe in detail our method for generating a discrete train track $TT_{\rho}$ from the revealing pair $\rho$ representing the general element $v\in V_{n}$ mentioned above, and briefly describe how to model the lamination it carries. We build an example discrete train track and lamination using the element $\alpha$ and the tree pair $(A,B,\sigma)$ from the previous subsection. Finally, we discuss some of the utility of $TT_{\rho}$ and some facts about how $TT_{\rho}$ would change under basic operations applied to $v$ (conjugation, then a choice of representative revealing pair) or $\rho$ (Salazar’s rollings). In the next subsection, we describe a derived object, the flow graph, which carries less information than the train track (although flow graphs in general still contain enough information to answer many dynamics questions). In our current practice, we find discrete train tracks and flow graphs to be helpful for understanding dynamics, while the generating revealing pairs tend to be helpful for any involved computations and for specifying elements with appropriate desired dynamics.

Note that any such discrete train track $TT_{\rho}$ is a representative of an equivalence class of similar train tracks, where the equivalence class is determined by all the discrete train tracks for revealing pairs equivalent to $\rho$ up to Salazar’s rollings, choice of location for drawing the complementary trees along the orbits of repelling and attracting fixed points, and conjugacy in $V_{n}$. Thus we are picking a representative object which is in some sense less well chosen than some “minimal” train track in this class. The result of applying Belk and Matucci’s geometric conjugacy invariant, the closed strand diagram, is very close to what a “minimal” discrete train track for an element of $V_{n}$ should be (leaving out the demarcations representing iterating the element, and of course, including some simplifications which result from the effects of conjugation (see [3])).

Here is how one draws a discrete train track from a revealing pair.

1. 1.

   List all of the iterated augmentation chains for the revealing pair.

2. 2.

   For each chain representing a non-trivial orbit of a periodic neutral leaf:

   1. (a)

      Draw a circle.

   2. (b)

      If the chain represents an orbit of length $r$, demarcate the circle into $r$ equal subintervals (typically we end these intervals with dots), and label these with the names of the nodes carrying the neutral leaves, in a clockwise order.

3. 3.

   For each chain of a repeller:

   1. (a)

      Draw a circle.

   2. (b)

      If the repelling periodic point travels an orbit of length $r$, then demarcate the circle into $r$ equal subintervals as above, and label these with the names of the nodes carrying the orbit of the repeller in a counter-clockwise order.

   3. (c)

      For the segment corresponding to the interval of the repelling periodic point of the complementary tree, instead of the one label mentioned in the last point, we label the two ends of the segment with the top and bottom of the spine of the repeller (top (root node of complementary component) before bottom in counter-clockwise order).

   4. (d)

      Lay the complementary component of the repeller along the sub-arc of the circle with the spine labels, gluing the spine to the circle.

      1. i.

         Scale the complementary component so that the spine is the length of the appropriate sub-arc of the circle.

      2. ii.

         Smooth out the tree (and the spine in particular) and bend the spine so that the spine has the same shape as the sub-arc of the circle with the labels from the spine nodes (preserving the current length of the spine).

      3. iii.

         Rigidly rotate the scaled, smoothed, and bent tree in the plane, and translate it so that the spine can be identified with the appropriate sub-arc of the circle.

4. 4.

   For each chain for an attractor:

   1. (a)

      Draw a circle.

   2. (b)

      If the attracting periodic point travels an orbit of length $r$, then demarcate the circle into $r$ equal subintervals, and label these with the names of the nodes carrying the orbit of the attractor in a counter-clockwise order.

   3. (c)

      For the segment corresponding to the interval of the attracting periodic point of the complementary tree, instead of the one label mentioned in the last point, we label the two ends of the segment with the top and bottom of the spine of the attractor (bottom (attracting leaf node of complementary component) before top in counter-clockwise order).

   4. (d)

      Lay the complementary component of the attractor along the sub-arc of the circle with the spine labels, gluing the spine to the circle.

      1. i.

         reflect the complementary component of the attractor across a vertical axis. Thus, the root of the component is now drawn at the bottom, while the left and right hand sides are preserved, respectively, as left and right hand sides.

      2. ii.

         Scale the complementary component so that the spine is the length of the appropriate sub-arc of the circle.

      3. iii.

         Smooth out the tree (and the spine in particular) and bend the spine so that the spine has the same shape as the sub-arc of the circle with the labels from the spine nodes (preserving the current length of the spine).

      4. iv.

         Rigidly rotate the reflected, scaled, smoothed, and bent tree in the plane, and translate it so that the spine can be identified with the appropriate sub-arc of the circle.

5. 5.

   For each source-sink chain, drawn a line connecting the appropriate sources and sinks (lines may have to cross each other in the case of $V_{n}$ which is why the lamination carried by the train track can only be embedded in an surface with boundary; strips can pass “under” each other). Demarcate each line with dots representing the length of the source-sink chain, and label sub-arcs with appropriate node labels as above for other sorts of iterated augmentation chains.

6. 6.

   Add parenthetical labels for splittings of trees where the support of the whole tree will be mapped away by one application of the element. Add parenthetical labels anywhere else as desired to improve clarity. (This step is not strictly necessary, but we find it to be helpful.)

If we follow the process above for the element $\alpha$, the diagram below is an example of what we may obtain (we include some drawn under-crossings following the methods from drawing knots from knot theory).

To visualize the lamination carried by a discrete train track, for each sub-arc in the demarcations of the discrete train tracks, one can draw a transverse copy of an $I$-fiber with an embedded copy of the Cantor set $\mathfrak{C}_{n}$. If we stabilize this by building a product with the interval $I$ (which lines we draw parallel to any train track sub-arc), then one has a picture of the lamination carried by the train track. (Glue the ends of the $I$-fibers of the Cantor sets using the rules provided by the map $v$, as a local picture of a mapping cone on the $I$-fiber running transverse to any demarcation dot, allowing contraction and expansion in the transverse $I$-fibers near to any tree-splitting.

Note that by reflecting the attracting complementary components as directed, we are able to draw the resulting carried lamination on a compact orientable surface with boundary. (The “left” and “right” portions of a Cantor set underlying a node are correctly associated without any twisting.)

Separately, one can observe that the group homomorphism $\mathcal{S}$ of the penultimate subsection of this article is connected with holonomy measurements for the carried lamination along repelling cycles when applied to the element we are centralizing.

A local diagram of the lamination carried by this train track local to the repelling cycle in the upper left corner is included in the diagram below. One can see the rescaling near the periodic repeller and portions of the Cantor set moving away along flow lines.

Let $TT_{1}$ be a discrete train track drawn in accordance with the method above. We define the support of the discrete train track $TT_{1}$ to be given as the union of the Cantor sets underlying the node labels in the iterated augmentation chains of the (non-trivial) periodic orbits, the source-sink chains, and the orbits of the repellers and attractors used to create the drawing of $TT_{1}$. Seeing a discrete train track as a (possibly disconnected) graph, we define the components of $TT_{1}$ to be the connected components of $TT_{1}$ as a graph. Given any such component $X$, we describe the support of the component $X$ to be the union of the Cantor sets underlying the node labels of $X$. We call a component a torsion component if the component is a circle generated from a periodic orbit of neutral leaves, otherwise we call a component a non-torsion component. Similar language will be defined for flow graphs, below.

We now mention some technical facts to do with the relationships amongst discrete train tracks drawn from distinct revealing pair representatives for an element of $V_{n}$, and also relationships which may arise as a consequence of conjugacy.

We give the basic definitions of Salazar’s rollings below, with the intention of understanding of the variance of discrete train tracks across the set of all representative revealing pairs for an element of $V_{n}$. The reader is encouraged to read Salazar’s Section 3.5 in [21], where she defines rollings of various types and traces the various impacts of rollings on revealing tree pairs.

The tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})$ is a single rolling of type E from $(A,B,\sigma)$ if it is obtained from $(A,B,\sigma)$ by adding an $n$-caret to $A$ and $B$ along each of the leaves of an iterated augmentation chain corresponding to one of two types of iterated augmentation chains; either to all of the leaves in $A$ and $B$ of a periodic orbit of neutral leaves, or to the initial source leaf in $A$ for a source-sink chain, and then to all of the neutral leaves in $A$ and $B$ of that chain, and then to the leaf of $B$ corresponding to the sink of that chain. (These are called elementary rollings.)

The tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})$ is a single rolling of type I from $(A,B,\sigma)$ if it is obtained from $(A,B,\sigma)$ by adding a cancelling tree along all of the leaves of $A$ along the orbit of a repeller, and at $B$ at the image of these leaves under the map, or by adding a cancelling tree at all the leaves of $A$ which appear in the reverse orbit of an attractor, and at the leaves of $B$ to which these leaves of $A$ are mapped. If $W$ is the complementary component of $A-B$ or $B-A$ corresponding to the repeller or attractor in this discussion, and $W$ is rooted at node $\Sigma$, then a tree $C$ is a cancelling tree for $W$ if it is obtained from $W$ by first choosing a proper, non-empty prefix $\Delta$ of the spine $\Gamma$ of $W$ (so that $\Gamma=\Delta\Theta$ for some suffix $\Theta$), and then taking $C$ to be the maximal sub-tree of $W$ which has root $\Sigma$ and containing the node $\Sigma\Delta$ as a leaf. (Note that if one carries out this process, the corresponding complementary component $W^{\prime}$ created in $A^{\prime}-B^{\prime}$ or $B^{\prime}-A^{\prime}$ for the tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})$ will now be rooted at $\Sigma\Delta$ and will have spine $\Theta\Delta$).

Finally, recall from sub-section 4.1 that a tree pair $(A^{\prime},B^{\prime},\sigma^{\prime})$ is a single rolling of type II from $(A,B,\sigma)$ if it is obtained from $(A,B,\sigma)$ by adding a copy of a component $U$ of $A-B$ to $A$ at the last leaf in the orbit of the repeller in $U$ and to $B$ at its image; or, by adding a copy of a component $W$ of $B-A$ to $A$ at the first leaf in the orbit of the attractor (the leaf of $A$ corresponding to the root node of $W$ in $B$) and to $B$ at its image.

The following lemma lists some basic properties of discrete train tracks. The reader will not be required to use this lemma later in the paper, although it gives a separate view of some arguments.

Proof: The latter points about rollings follow directly from the definitions of rollings, as given in subsection 4.1.

The first two sub-points of 1 are a result of the fact that $V_{n}$ is a group of homeomorphisms, and topological conjugacy preserves the properties mentioned. The third sub-point of 1 follows from the fact that elements of $V_{n}$ do not change infinite suffixes; so, the points in the finite orbit of a repelling periodic point or of an attracting periodic point all have the same infinite repeating suffix (that is, as described in Lemmas 4.3 and 4.2). Now the conjugating element $f$ again cannot change this infinite suffix class, so the resulting orbit will consist of points with this same infinite suffix. (Note first that that Belk and Matucci [3] also derive the infinite suffix of a finite repelling or attracting orbit as a conjugacy invariant for elements of $F<T<V$, using a method very similar to our discrete train tracks, although Belk and Matucci’s definition is mildly different, and second that one can choose a representative revealing pair for the conjugate version $\tau_{2}$ of $\tau_{1}$ so that the spine is cyclically rotated, but by applying a rolling of type $II$, this spine can be rotated back to the original spine, thus producing a discrete train track with the same spine labelling on the appropriate sub-arc of the circle (although, the prefixes of all nodes on the circle will be lengthened).)   $\diamond$