arXiv:1609.00771v2 [math.DS] 28 Sep 2016

ROTATION NUMBERS OF ELEMENTS IN THOMPSON’S GROUP T
JEFFREY DILLER AND JAN-LI LIN
Abstract. We give a simple combinatorial proof that the rotation number for each element in
Thompson’s group T is rational.

1. Introduction
In 1965, Richard Thompson defined three groups which have furnished counterexamples to various conjectures in group theory. One of these T, which will be the subject of this article, is the
group of ‘dyadic’ circle homeomorphisms f : S1 → S1 . That is, if one takes S1 to be the interval
[0, 1] with endpoints identified, then
• f preserves the set of dyadic rational numbers (i.e., numbers of the form p · 2q , p, q ∈ Z),
• f is linear except at a finite number of dyadic rational points,
• on each interval such that f is linear, the derivative (i.e., slope) is a power of 2.
For more information on Thompson’s groups, see [CFP,G,GS]. (Notice that the group T is denoted
by G in [G, GS].)
The rotation number [KH] of an orientation preserving circle homeomorphism f : S1 → S1 is the
quantity
fen (x)
mod Z,
ρ(f ) = lim
n→∞
n
where fe : R → R is any lift of f to the universal cover R, and x ∈ R is any initial point. Modulo
Z, the limit does not depend on the choices of fe and x, and it is an important dynamical invariant
for f . In particular ρ(f ) is rational if and only if f has a periodic point. In 1987, E. Ghys and V.
Sergiescu [GS] proved
Theorem A.
I. For every element f in Thompson’s group T, the rotation number ρ(f ) is rational.
II. For every rational number r, there is an element f ∈ T such that ρ(f ) = r mod Z.
However, as Ghys pointed out in [G], their proof is very indirect, and there is a need for a better
proof. Several simpler proofs were given later, in articles by Liousse [L] and Calegari [C] and
the thesis of Matucci [M]. An approach via revealing pairs, implicit in the article [BBGL+ ], was
discussed at length during a 2007 Luminy talk by Bleak. The goal of this paper is to provide another
elementary combinatorial proof of their result. It is similar in many respects to those of Calegari and
Matucci, but ours uses a different description of T as the group of piecewise linear automorphisms
of Z2 , and where Calegari uses Thurston’s notion of a ‘train track’ ours uses combinatorial ideas
1991 Mathematics Subject Classification. 37E10, 37E45.
Key words and phrases. Thompson’s group, rotation number, fans, cones.
This work was supported in part by National Science Foundation grant DMS–1066978.
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JEFFREY DILLER AND JAN-LI LIN

connected with ‘fans’, i.e. partitions of Z2 ⊂ R2 into rational convex cones. Like his proof, ours
provides an effective way to compute rotation numbers for elements in T.
Fans in R2 arise naturally in the study of toric surfaces, and the elements of T arise in particular
(see [U, F]) as “tropicalizations” of plane birational maps preserving the (C∗ )2 -invariant two form
dx∧dy
xy . The ideas in this paper descend from our recent work [DL, Theorem E] concerning dynamics
of such maps, and an older result [DF, Theorem 0.1] of Favre and the first author. Nevertheless,
the presentation here is purely combinatorial, based on an analysis of how elements of T act on
fans and their refinements, and it makes no appeal to algebraic geometry.
The rest of the paper proceeds as follows. We introduce terminology and notation associated to
cones, fans, and piecewise linear automorphisms of Z2 in Section 2, concluding with a description
of the relationship between piecewise linear automorphisms and dyadic circle homeomorphisms. In
Section 3 we establish a key result (Proposition 3.3) about decomposing piecewise linear automorphisms. Section 4 then concludes the proof of Theorem A.
We are indebted to Charles Favre for pointing out that while the Theorem A is a corollary
of results in our earlier paper [DL], one can extract from that source a direct proof involving no
algebraic geometry. We would also like to thank Victor Kleptsyn for informing us about the papers
[L] and (especially) [C] mentioned above. Finally, we would like to thank the referee for a careful
reading and for correcting many notational inconsistencies and typos.
2. Piecewise Linear Automorphisms of Z2
Let R+ := {x ∈ R | x ≥ 0} denote the set of non-negative real numbers. A ray in R2 is a rational
one dimensional cone, i.e. a set of the form τ = R+ v for some v ∈ Z2 \ {0}. The first lattice
point v ∈ τ ∩ (Z2 \ {0}) is called the generator of τ . We will call one dimensional cones with
(possibly) irrational slope real rays. A sector is a rational two dimensional cone, i.e. a convex set
σ = R+ v1 + R+ v2 for some v1 , v2 ∈ Z2 \ {0} linearly independent over R. The rays τi = R+ vi ,
i = 1, 2, bounding σ are called facets of σ. The sector σ is regular if the generators v1 , v2 of the
facets of σ form a basis for Z2 , i.e. if | det[v1 v2 ]| = 1.
A fan ∆ is a set of cones consisting of the zero-dimensional cone {0}, a finite sequence of rays
τ0 , τ1 , · · · , τd given in counterclockwise order, and the intervening sectors σj bounded by τj−1 and
τj , 1 ≤ j ≤ d. The support of ∆, denoted supp(∆), is the union of all cones in ∆. We assume
throughout that our fans are complete, i.e. i.e., the sectors σj in ∆ cover R2 (hence τ0 = τd ). A
fan is regular if all its sectors are regular.
A continuous map F : R2 → R2 is piecewise linear if there exists a fan ∆ with sectors σ1 , · · · , σd
and linear transformations Li : R2 → R2 , i = 1, · · · , d such that F |σi = Li |σi for all i = 1, · · · , d.
Any fan satisfying the above condition is said to be compatible with F . The map F is orientation
preserving if and only if each of Li is. A piecewise linear automorphism of Z2 is an orientation
preserving, piecewise linear homeomorphism F : R2 → R2 such that F (Z2 ) = Z2 . If F is a
piecewise linear automorphism, then all the maps Li in the definition must have det(Li ) = 1, i.e.,
Li ∈ SL(2, Z). It follows that if ∆ is a regular fan compatible with F , then the fan F (∆) obtained
by mapping forward all cones in ∆ is also regular, though it is not necessarily compatible with F .
2.1. Equivalence between realizations of T. In proving Theorem A we will always think of
elements of T as piecewise linear automorphisms of Z2 . Nevertheless, since T is more commonly
given as the group of dyadic circle homeomorphisms, we digress briefly to indicate how to translate
between the two points of view. Every piecewise linear homeomorphism F : R2 → R2 induces a

ROTATION NUMBERS OF ELEMENTS IN THOMPSON’S GROUP T

3

circle homeomorphism by projectivizing. More precisely, F permutes the set R of real rays in R2 ,
which are parameterized by S1 . If, moreover, F restricts to a bijection Z2 → Z2 , then the induced
homeomorphism of S1 completely determines F . It remains to give a homeomorphism φ : R → S1
that conjugates (the projectivizations of) piecewise linear automorphisms of Z2 to dyadic circle
homeomorphisms. In the following, we will regard S1 as the interval [0, 1] with endpoints identified
and define a map φ : R → [0, 1].
] be a “dyadic standard1” interval and
This is accomplished locally as follows. Let I = [ 2ak , a+1
2k
σ ⊂ R2 be a regular sector generated by the (oriented) basis v1 , v2 ⊂ Z2 . Taking Rσ ⊂ R to be the
set of real rays in σ, we define a homeomorphism φ : Rσ → I as follows, identifying each rational
ray with its generator v ∈ Z2 . We first assign facets to endpoints: φ(v1 ) = 2ak and φ(v2 ) = a+1
.
2k
1
Then we proceed inductively by averaging: φ(v1 + v2 ) = 2 (φ(v1 ) + φ(v2 )), and more generally, if
φ is defined at two primitive points (i.e. integral points where the two coordinates are coprime)
v, v ′ ∈ σ ∩ Z2 but at no points in the intervening sector, then we set φ(v + v ′ ) = 12 (φ(v) + φ(v ′ )).
One checks that this procedure results in an order-preserving bijection between rational rays in
σ and dyadic points in I which therefore extends to an orientation-preserving homeomorphism
φ : Rσ → I. Note that when I = [0, 1] and σ is the cone generated by (1, 0) and (1, 1) ∈ R2 with
Rσ parameterized by the slope, φ is known as Minkowski’s question mark function.
One globalizes φ by choosing any partition R2 = σ1 ∪ · · · ∪ σd into regular sectors and a corresponding partition [0, 1] = I1 ∪ · · · ∪ Id into standard dyadic intervals and defining φ|Rσj : Rσj → Ij
sector-wise as above.
To see that φ conjugates piecewise linear automorphisms of Z2 to dyadic circle maps, it suffices
to verify the following additional fact. Let φ : Rσ → I be the ‘local’ version of φ defined above. Let
σ ′ be another regular sector, I ′ another standard dyadic interval, and φ′ : Rσ′ → I ′ the analogous
homeomorphism. Then φ′ ◦F = f ◦φ where F ∈ SL(2, Z) is the unique element such that F (σ) = σ ′
and f : R → R is the unique increasing affine map such that f (I) = I ′ .
3. Decomposition of piecewise linear automorphisms of Z2
A refinement of a fan ∆ is a fan ∆′ with the same support as ∆ and such that for every cone
∈ ∆′ , there exists σ ∈ ∆ such that σ ′ ⊆ σ. The following lemmas are well-known in convex
geometry, but we include the proofs for completeness.

σ′

Lemma 3.1. Any fan admits a regular refinement.
Proof. It suffices to prove that any sector σ may be partitioned into regular sectors. If σ is not
regular, then the ray generators u1 , u2 of its facets must satisfy | det[u1 u2 ]| ≥ 2. This implies that
there must be at least one primitive point w in the interior of the parallelogram formed by u1 and
u2 . The ray R+ w divides σ into two new sectors. Since | det[w ui ]| < | det[u1 u2 ]|, i = 1, 2, we see
that finitely many such subdivisions will result in a partition of σ into regular sectors.

Let σ ∈ ∆ be a regular sector with its boundary rays generated by u1 , u2 ∈ Z2 . Let τ ⊂ σ be
the ray generated by some point u ∈ Z2 in the interior of σ. Then σ \ τ is a union of two disjoint
sectors σ1 , σ2 . One checks that they are both regular if and only if u = u1 + u2 . We then call the
fan ∆′ = (∆ \ {σ}) ∪ {σ1 , σ2 , τ } the simple split of ∆ at σ. If ∆ is regular then so is ∆′ . Conversely,
we call ∆ the simple merge of ∆′ at σ1 and σ2 .
1A dyadic sub-interval I ⊂ [0, 1] is called standard if it is of the form I = [ a ,
2k

[ 14 , 12 ]

is dyadic and standard, whereas

[ 14 , 43 ]

is dyadic but not standard.

a+1
].
2k

For example, the interval

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JEFFREY DILLER AND JAN-LI LIN

Lemma 3.2. Suppose that ∆, ∆′ are regular fans, and ∆′ refines ∆. Then one can obtain ∆′ from
∆ by a sequence of simple splits.
Proof. We need to consider only the case where ∆ is the fan determined by a single regular sector
σ := R+ u + R+ v. Working inductively, it suffices to show that either ∆′ includes the ray τ :=
R+ (u + v) that barycentrically subdivies σ, or ∆′ = ∆. If τ ∈
/ ∆′ , then we let τ1 = R+ (au + bv),
′
′
τ2 = R+ (cu+dv) be the facets of the sector σ ∈ ∆ that contains τ . We may assume that a > b ≥ 0
and d > c ≥ 0 are all integers. Since σ ′ is regular, we get
1 = ad − bc ≥ (b + 1)(c + 1) − bc = b + c + 1.
Thus b = c = 0 and a = d = 1, i.e. σ ′ = σ and ∆′ = ∆.



From now on, F will always denote a piecewise linear automorphism of Z2 . Our proof of Theorem
A amounts to carefully analyzing how fans transform under F . Note that if ∆ is a regular fan
compatible with F , then the image fan F (∆) := {F (σ) : σ ∈ ∆} is also regular. If ∆′ is another
regular fan, then we say that F is an isomorphism from ∆ to ∆′ if F (∆) = ∆′ . Similarly, we say
that F is a simple split or simple merge of ∆′ if F (∆) is accordingly, a simple split or simple merge
of ∆′ . We call F a simple map from ∆ to ∆′ if F is of one of these three types.
Regardless, we associate to F a partially defined, ‘approximate’ map F♯ : ∆ → ∆′ as follows: for
each σ ∈ ∆, we take F♯ σ to be the smallest cone in ∆′ containing F (σ). If no single cone in ∆′
contains F (σ), then we leave F♯ σ undefined. Hence F♯ τ is defined for all rays τ ∈ ∆, though the
image might be a sector or a ray; whereas for each sector σ ∈ ∆, the cone F♯ σ is either another
sector or undefined. If G is another piecewise linear automorphism of Z2 and G♯ : ∆′ → ∆′′ is an
approximate map of fans, then we have G♯ F♯ σ = (G ◦ F )♯ σ provided that the left side is defined.
Proposition 3.3. Let ∆ be a regular fan compatible with F . Then there is a decomposition F =
fn−1 ◦ · · · ◦ f0 into piecewise linear automorphisms of Z2 and a corresponding sequence of regular
fans ∆0 , . . . , ∆n such that
• ∆0 = ∆n = ∆;
• ∆j is compatible with fj for 0 ≤ j ≤ n − 1;
• fj is a simple map from ∆j to ∆j+1 .
Proof. First, set f0 = F and ∆1 = F (∆). Lemma 3.1 then gives a regular common refinement ∆′
of ∆ and F (∆). Then by Lemma 3.2, we know that we can obtain ∆′ from F (∆) by performing
finitely many simple splits, i.e. we have a sequence
∆

F

/ ∆1 = F (∆)

id

/ ∆2

id

/ ···

id

/ ∆k = ∆′

in which all but the first arrow represents a simple split; and the first arrow is an isomorphism.
Similarly, there is a sequence
∆k = ∆′

id

/ ∆k+1

id

/ ···

id

/ ∆n−1

id

/ ∆ = ∆n

such that each arrow is a simple merge, i = k, · · · , n − 1. Putting the two sequences together
completes the proof.


ROTATION NUMBERS OF ELEMENTS IN THOMPSON’S GROUP T

5

4. Proof of Theorem A
We call a cone τ ∈ ∆ deterministic for the approximate self-map F♯ : ∆ → ∆ if all its approximate
forward images (F♯ )j τ , j ≥ 1, are defined.
Proposition 4.1. For any piecewise linear automorphism F and any fan ∆, there exists a regular refinement ∆′ of ∆ such that ∆′ is regular and compatible with F , and every ray in ∆′ is
deterministic for F♯ : ∆′ → ∆′ .
Proof. Let fj and ∆j be the maps and associated fans in the decomposition of F from Proposition
3.3. Since ∆0 = ∆n , we may extend these sequences of maps/fans periodically, e.g. fj := fj mod n
for all j ∈ N. Suppose that there exists a ray τ ∈ ∆ that is not deterministic for F♯ : ∆ → ∆. In
particular, we know F (∆) 6= ∆, so n ≥ 2. Then by replacing τ with an approximate forward image,
we may assume that F♯ τ = σ is a sector and that (F♯ )k σ is undefined for k ∈ N large enough.
We may refine this observation using the decomposition of F . Namely, there exist indices i <
k ∈ N, and a ray τi ∈ ∆i such that
(i) σi+1 := fi♯ τi ∈ ∆i+1 is a sector;
(ii) σj+1 := fj♯σj ∈ ∆j+1 is well-defined for j = i + 1, · · · , k − 1, but fk♯σk is not defined;
(iii) i, τi , k are chosen so that k − i is as small as possible given the first two conditions.
That is, we attend to the following sequence:
τi ∈ ∆i

fi

/ σi+1 ∈ ∆i+1

fi+1

/ ···

fk−1

/ σk ∈ ∆ k

fk

/ ∆k+1

The idea is to inductively refine the fans ∆i+1 , · · · , ∆k so that fk becomes an isomorphism.
Condition (i) together with the fact that fi is simple from ∆i to ∆i+1 means that τi+1 := fi (τi )
is the unique ray that divides σi+1 into two regular cones. Replacing σi+1 with these two cones and
adding the ray τi+1 to ∆i+1 turns the map fi : ∆i → ∆i+1 into an isomorphism from ∆i to ∆i+1 .
Note that in this (and succeeding) steps, we apply the same refinement to all fans ∆i+ℓn equivalent
mod n to ∆i . Since n ≥ 2 this does not affect the outcome.
Turning from fi to fi+1 , we find two possible cases. If i + 1 = k, then fi+1 (σi+1 ) is equal to
the union of two adjacent sectors in ∆i+2 . Since the above refinement of ∆i+1 split σi+1 into two
regular cones, fi+1 is now an isomorphism from ∆i+1 to ∆i+2 .
Otherwise, i + 1 < k. If fi+1 was initially a merge, then there is another ray τ̃i+1 ∈ ∆i+1 such
that fi+1 (τ̃i+1 ) ∈
/ ∆i+2 . However, our minimality assumption on k − i guarantees that the images
fi+1 (τ̃i+1 ) and fi+1 (τi+1 ) lie in different sectors. Hence regardless of whether fi+1 is a merge or not,
fi+1 (τi+1 ) divides σi+2 into two regular cones. So as with ∆i+1 , we can refine ∆i+2 by splitting
σi+2 into two regular sectors, the result being that fi+1 remains a simple map. In this case, we
move on to fi+2 and repeat the process until we reach fk . In the end, fk is improved from a split
to an isomorphism, and maps fj with j 6≡ k mod n remain simple of the same type as before.
If there remain rays in ∆ = ∆0 that are not deterministic for F♯ , then we repeat the process.
Since there are only finitely many fj (modulo n), finitely many repetitions leads to a situation in
which either all rays in ∆ are deterministic or all fj are isomorphisms. In the latter case F is itself
an isomorphism from ∆ to ∆, and all rays are deterministic anyhow.

Finally, we can now prove the first part of Theorem A.
Proof of Theorem A, part I. Given any piecewise linear automorphism F of Z2 , we first find any
fan ∆ such that ∆ is compatible with F . Next, we find a regular refinement ∆′ of ∆, then ∆′

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JEFFREY DILLER AND JAN-LI LIN

is still compatible with F . Finally, we find a further refinement ∆′′ such that every ray in ∆′′ is
deterministic for F♯ . Observe how F acts on the rays and sectors of ∆′′ , either one of the following
will happen:
• There is never a ray mapped into a sector, so rays always map to rays, and F is permuting
the rays. In this case, some iterate of every ray is going back to itself and F is of finite
order.
• There is a ray maps into a sector σ, then σ must be deterministic for F♯ , and (F♯ )j σ, j ≥ 1,
are all defined and are sectors. Then again, we have finitely many sectors, there must be
some i < j such that (F♯ )i σ = (F♯ )j σ. As a consequence, there is a ray in (F♯ )i σ that is
fixed by F j−i , so the circle map induced by F has a periodic point.
In either case, the rotation number of F is rational because there is a periodic point. This
completes the proof.

Remark 1. Our proof also gives an algorithm for finding the rotation number of a piecewise linear
automorphism F . First, find a fan ∆ that is compatible with F . Then find a regular refinement ∆′
of ∆ such that every ray in ∆′ is deterministic for F♯ . This can be done effectively by Lemma 3.1,
Lemma 3.2, and Proposition 3.3. Tracing the orbit of each ray in ∆′ , as we did in the above proof,
will locate the sectors containing periodic (real) rays, and this will then give the rotation number
of F .
Remark 2. We stress here that in the definition of piecewise linear automorphism, the condition
2
2
F (Z2 ) = Z2 is crucial for the Theorem
 4 −3A to hold. If we only require that F (Z ) ⊆ Z , then it is
easy to find a linear map, e.g., F = 3 4 such that the corresponding circle homeomorphism has
an irrational rotation number.
The second part of Theorem A can be proved by a concrete construction.
Proof of Theorem A, part II. It suffices to show that for any given integer n ≥ 3, we can find a
piecewise linear automorphism F of Z2 such that ρ(F ) = 1/n. For each integer n ≥ 3 we can find
a regular fan ∆ with n sectors, say σ0 , σ1 , · · · , σn = σ0 , arranged in the counterclockwise order.
Since each σi is regular, there is a unique Li ∈ SL2 (Z) such that Li maps σi bijectively onto σi+1 ,
i = 0, · · · , n − 1 and maps the lattice points in σi bijectively onto lattice points in σi+1 . Define F
as F |σi = Li |σi , i.e., F is rotating the sectors, sending σi to σi+1 . Then F is periodic of period n,
and therefore ρ(F ) = 1/n.

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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
E-mail address: diller.1@nd.edu
Department of Mathematics, Northwestern University, Evanston, IL 60208
E-mail address: janlin@math.northwestern.edu

