Rotation Sets

Abstract

Throughout this chapter d will be a fixed integer ≥ 2. We study certain invariant sets for the multiplication by d map $m_d: {\mathbb T} \to {\mathbb T}$ defined by

$$\displaystyle m_d(t) = d t \qquad (\operatorname {mod} \ {\mathbb Z}). $$

The low-degree cases m ₂ and m ₃ are often referred to as the doubling and tripling maps.

Throughout this chapter d will be a fixed integer ≥ 2. We study certain invariant sets for the multiplication by d map $m_d: {\mathbb T} \to {\mathbb T}$ defined by

$$\displaystyle \begin{aligned} m_d(t) = d t \qquad (\operatorname{mod} \ {\mathbb Z}). \end{aligned} $$

The low-degree cases m ₂ and m ₃ are often referred to as the doubling and tripling maps.

FormalPara Definition 2.1

A non-empty compact set $X \subset {\mathbb T}$ is called a rotation set for m _d if

• X is m _d -invariant in the sense that m _d (X) = X,¹ and

• the restriction m _d | _X can be extended to a degree 1 monotone map of the circle.

Roughly speaking, the latter condition means that m _d preserves the cyclic order of all triples in X, except that it may identify some pairs.

If X is a rotation set for m _d and g, h are degree 1 monotone extensions of m _d | _X , then g = h on every orbit in X, so ρ(g) = ρ(h) by Theorem 1.8. This quantity, which therefore depends on X only, is called the rotation number of X and is denoted by ρ(X). We refer to X as a rational or irrational rotation set according as ρ(X) is rational or irrational.

2.1 Basic Properties

Since multiplication by d commutes with the rigid rotation r : t↦t + 1∕(d − 1) $(\operatorname {mod} \ {\mathbb Z})$, if X is a rotation set for m _d , so are its d − 2 rotated copies

$$\displaystyle \begin{aligned} X+\frac{1}{d-1}, \ X+\frac{2}{d-1}, \ \ldots, \ X+\frac{d-2}{d-1} \quad (\operatorname{mod} \ {\mathbb Z}). \end{aligned}$$

Moreover, all these sets have rotation number ρ(X) since if g is a monotone extension of m _d | _X , then the conjugate map r ^∘i ∘ g ∘ r ⁻ⁱ will be a monotone extension of the restriction of m _d to X + i∕(d − 1).

Example 2.2

The 3-cycle $X: \frac {7}{26} \mapsto \frac {21}{26} \mapsto \frac {11}{26}$ under tripling is a rotation set with rotation number $\frac {2}{3}$. Two possible monotone extensions of m ₃ restricted to this cycle are shown in Fig. 2.1. The 180^∘-rotation of X produces the new rotation set $X+\frac {1}{2}: \frac {8}{26} \mapsto \frac {24}{26} \mapsto \frac {20}{26}$ with the same rotation number. On the other hand, the 4-cycle $\frac {1}{5} \mapsto \frac {3}{5} \mapsto \frac {4}{5} \mapsto \frac {2}{5}$ under tripling is not a rotation set since it fails to have a combinatorial rotation number (compare Corollary 1.16).

Fig. 2.1

The cycle $X: \frac {7}{26} \mapsto \frac {21}{26} \mapsto \frac {11} {26}$ under tripling is a rotation set with $\rho (X)= \frac {2}{3}$. Left: A generic monotone extension of m ₃| _X . Right: The “standard” monotone extension of m ₃| _X (see the discussion leading to Eq. (2.1))

A rotation set containing periodic orbits is clearly rational. Conversely, every orbit in a rational rotation set is eventually periodic. Here is a more precise statement:

Theorem 2.3

Suppose X is a rational rotation set for m _d , with ρ(X) = p∕q in lowest terms. Then, every forward orbit in X under m _d is finite. More precisely, for every t ∈ X there is an integer i ≥ 0 such that $m_d^{\circ i}(t)$ is periodic of period q. In particular, X is at most countable.

Proof

Take any t ∈ X and any degree 1 monotone extension g of m _d | _X . We know from Theorem 1.14 that the sequence $\{ g^{\circ nq}(t) = m_{d^q}^{\circ n}(t) \}$ tends to a periodic point t′∈ X of period q as n →∞. Since the map $m_{d^q}$ is uniformly expanding on the circle, its fixed point t′ is repelling. Hence $m_{d^q}^{\circ n}(t)$ cannot converge to t′ unless $m_{d^q}^{\circ n}(t) = t'$ for some n. □

Remark 2.4

Most periodic orbits of m _d do not define rotation sets. For each prime number q the equation m ^∘q(t) = t has d ^q − 1 solutions $t = i/(d^q-1) \, (\operatorname {mod} \ {\mathbb Z})$. Discarding the d − 1 fixed points of m _d , it follows that the number (d ^q − d)∕q of distinct q-cycles of m _d grows exponentially fast as q →∞. On the other hand, the number of q-cycles of m _d that form a rotation set is precisely $(q-1) \ {q+d-2 \choose q}$, which grows like the power q ^d−1 as q →∞ (see Corollary 3.11).

Every rotation set is nowhere dense since any open interval on the circle eventually maps to the whole circle under the iterations of m _d . By contrast, Lebesgue measure on the circle is ergodic for m _d ,² so a randomly chosen point on ${\mathbb T}$ has a dense orbit almost surely. This proves the following

Theorem 2.5

The union ${\mathcal R}_d$ of all rotation sets for m _d has Lebesgue measure zero.

McMullen [19] has proved the sharper statement that the Hausdorff dimension of ${\mathcal R}_d$ is zero.³ For more on the set ${\mathcal R}_d$, see Sect. 4.3.

To study of the structure of a rotation set, we first look at its complement.

Definition 2.6

Let X be a rotation set for m _d . A connected component of the complement ${\mathbb T} \smallsetminus X$ is called a gap of X. A gap of length ℓ is minor if ℓ < 1∕d and major otherwise. The multiplicity of a major gap is the integer part of dℓ ≥ 1. A major gap is taut or loose according as dℓ is or is not an integer.

Intuitively, a minor gap is short enough so it maps homeomorphically onto its image by m _d . On the other hand, a major gap is too long and wraps around the circle by m _d as many times as its multiplicity (see Lemma 2.8 below).

It will be convenient to work with a specific degree 1 monotone extension of m _d | _X which can be defined whenever X has more than one point. This map, which we call the standard monotone map of X and denote by g _X , is defined as follows: On every minor gap, set g _X = m _d . On every major gap (a, a + ℓ) of length 0 < ℓ < 1 and multiplicity n, define

$$\displaystyle \begin{aligned} g_X(t)= \begin{cases} m_d(a) & t \in \Big( a, a+\frac{n}{d} \Big] \vspace{2mm} \\ m_d(t) & t \in \Big( a+\frac{n}{d}, a+\ell \Big) \end{cases} \end{aligned} $$

(2.1)

(see Figs. 2.1 and 2.2). The map g _X is piecewise affine with derivatives 0 and d, so the total length of the plateaus of g _X is 1 − 1∕d = (d − 1)∕d. Since by the construction each major gap of multiplicity n contributes a plateau of length n∕d, we arrive at the following fundamental fact (compare [2] and [4]):

Fig. 2.2

Left: The standard monotone map g _X of some rotation set X for m ₅. Counting multiplicities, X has four major gaps, two taut gaps I ₁, I ₃ of multiplicity 1 and a loose gap I ₂ of multiplicity 2. Right: The position of major gaps around the circle. Notice that each major gap contains as many fixed points of m _d as its multiplicity, as asserted in Lemma 2.13

Theorem 2.7

Every rotation set for m _d containing more than one point has d − 1 major gaps counting multiplicities.

The following lemma summarizes the mapping properties of gaps:

Lemma 2.8

Let X be a rotation set for m _d containing more than one point and I = (a, a + ℓ) be a gap of X. Take any degree 1 monotone extension g of m _d | _X .

• (i) If I is a minor gap, the interior J of g(I) is a gap of length dℓ. Moreover, m _d  : I → J is a homeomorphism.

• (ii) If I is a taut gap of multiplicity n, the image g(I) is the single point m _d (a) ∈ X. Under m _d , the point m _d (a) has n − 1 preimages in I, whereas every point in ${\mathbb T} \smallsetminus \{ m_d(a) \}$ has n preimages in I.

• (iii) If I is a loose gap of multiplicity n, the interior J of g(I) is a gap of length dℓ − n. Under m _d , every point in J has n + 1 preimages in I, whereas every point outside J has n preimages in I.

Proof

For the standard monotone map g _X the statements follow immediately from the definition. For an arbitrary extension g, we can use the fact that g is monotone and takes the same values as g _X on the boundary of gaps to arrive at the same conclusions. The details are straightforward and will be left to the reader. □

The preceding lemma shows that the pattern of how gaps map around is independent of the choice of the monotone extension g. For any gap I, the image g(I) is either a point or a gap J modulo its endpoints. In practice, it is convenient to ignore the issue of endpoints and simply declare that I maps to J. With this convention in mind, we see from the above lemma that every minor gap eventually maps to a major gap I. If I is taut, it maps to a point and the process stops. If I is loose, it maps to a new gap and the process continues.

Let us collect some corollaries of this basic observation.

Theorem 2.9

A rotation set is uniquely determined by its major gaps.

Proof

Let X, Y  be rotation sets with the same collection of major gaps. We may assume neither of X, Y  is a single point. Suppose there is some $t \in Y \smallsetminus X$. Then t must belong to a minor gap I of X. Take the smallest integer i > 0 such that $J=m_d^{\circ i}(I)$ is a major gap of X. Then $m_d^{\circ i}:I \to J$ is a homeomorphism, so $m_d^{\circ i}(t) \in J \cap Y$, which is impossible since J is a major gap of Y  as well. This proves Y ⊂ X. Similarly, X ⊂ Y . □

Theorem 2.10

Suppose X is a rotation set containing more than one point and I is a gap of X. Then either I is periodic or it eventually maps to a taut gap.

Proof

Let I _i denote the interior of $g_X^{\circ i}(I)$ and assume that I _i is not taut for any i. By Lemma 2.8 there is a sequence i ₁ < i ₂ < i ₃ < ⋯ of positive integers for which $I_{i_k}$ is loose. Since there are finitely many loose gaps, we must have $I_{i_j}=I_{i_k}$ for some j < k. This proves that I eventually maps to a periodic gap. Since by monotonicity of g _X every gap is the image of precisely one gap, it follows that I itself must be periodic. □

Corollary 2.11

Every infinite rotation set has at least one taut gap.

Conversely, all major gaps of a finite rotation set are loose since in this case m _d , being surjective, must also be injective on the rotation set.

Proof

Otherwise every gap would be periodic by the previous theorem, so its endpoints would be periodic points in the rotation set. By Theorem 1.14 these infinitely many endpoints would have the same period q > 0 under m _d . This is impossible since m _d has only finitely many q-cycles. □

Remark 2.12

Here is an alternative approach to the above corollary (compare [2]): Lemma 2.8 applied to g _X shows that m _d (t) = m _d (t′) for a distinct pair t, t′∈ X precisely when t, t′ form the endpoints of a taut gap or more generally when there is a chain t = t ₁, t ₂, …, t _k  = t′∈ X such that each pair t _i , t _i+1 forms the endpoints of a taut gap. Thus, if X had no taut gap, the map m _d  : X → X would be a homeomorphism. Since m _d is expanding, this would imply that X is finite [21, Lemma 18.8].

The next result establishes a connection between the major gaps of a rotation set and the d − 1 fixed points

$$\displaystyle \begin{aligned} u_i=\frac{i}{d-1} \ \ (\operatorname{mod} \ {\mathbb Z}) \end{aligned}$$

of the map m _d . This connection will play an important role in Sects. 3.2 and 3.3.

Lemma 2.13

Suppose X is a rotation set for m _d with ρ(X) ≠ 0. Then each major gap of X of multiplicity n contains exactly n fixed points of m _d .

Compare Fig. 2.2.

Proof

The assumption ρ(X) ≠ 0 tells us that each fixed point of m _d belongs to a gap, which is necessarily major since a minor gap is disjoint from its image under m _d . Let I be a major gap of multiplicity n and assume that it contains n + 1 adjacent fixed points u _i , …, u _i+n. Since each open interval (u _j , u _j+1) contains precisely one preimage of every fixed point under m _d , it follows that u _i has at least n + 1 preimages in I. By Lemma 2.8, I is loose and u _i belongs to the interior of g _X (I). This implies that the closure of I maps onto itself by g _X , so the endpoints of I must be fixed by m _d , which contradicts the assumption ρ(X) ≠ 0. Thus, I contains at most n fixed points of m _d .

Now let {I _i } be the finite collection of major gaps of X of multiplicities {n _i }. We have shown that the number k _i of fixed points in I _i satisfies 0 ≤ k _i  ≤ n _i . Since $\sum k_i = \sum n_i =d-1$, we must have k _i  = n _i for all i. □

To each rotation set X for m _d we can assign a gap graph Γ _X which is a finite directed (not necessarily connected) graph having one vertex for each major gap of X, with an edge going from vertex I to vertex J whenever J is the first major gap in the forward orbit of I. We also assign to each vertex I a weight w(I) ≥ 1 equal to its multiplicity. Thus, Γ _X has the following properties:

• (i) $\displaystyle {\sum _{\text{vertices} \, I}} w(I) = d-1$.

• (ii) The degree of every vertex is either 0 (no edge going out or coming in), or 1 (only one edge going out or coming in), or 2 (one edge going out and one coming in, possibly a loop).

If X has no loose gaps, Γ _X is a trivial graph consisting of at most d − 1 vertices and no edges. If X is an irrational rotation set, Theorem 2.10 tells us that every directed path in Γ _X terminates at a taut vertex and in particular there are no closed paths (see Fig. 2.3).

Fig. 2.3

Possible gap graphs for irrational rotation sets under m _d for 2 ≤ d ≤ 5. The red and blue vertices correspond to taut and loose gaps respectively, and the weights denote multiplicities

Let us call a finite directed graph admissible of degree d if it satisfies the conditions (i) and (ii) above. It is natural to ask the following

Question 2.14

Given an admissible graph Γ of degree d, does there exist a rotation set X for m _d whose gap graph Γ _X is isomorphic to Γ?

In Sect. 4.2 we will provide the answer to this question in the case Γ has no closed paths (see Theorem 4.6).

2.2 Maximal Rotation Sets

Take any collection

$$\displaystyle \begin{aligned} \mathbb{I} = \{ I_1, \ldots, I_{d-1} \} \end{aligned}$$

of disjoint open intervals on the circle, each of length 1∕d. Consider the set

$$\displaystyle \begin{aligned} X_{\mathbb{I}} = \{ t \in {\mathbb T}: m_d^{\circ n}(t) \notin I_1 \cup \cdots \cup I_{d-1} \ \text{for all}\ n \geq 0 \}. \end{aligned}$$

Theorem 2.15 ([4])

$X_{\mathbb {I}}$ is a rotation set for m _d .

Proof

First we check that $X_{\mathbb {I}} \neq \emptyset$. Denote by U the open set I ₁ ∪⋯ ∪ I _d−1. Under m _d , every $t_0 \in {\mathbb T}$ has d preimages which are a distance 1∕d apart, hence at least one of these preimages, say t ₁, must be outside U. It follows inductively that there is a backward orbit ⋯↦t ₂↦t ₁↦t ₀ such that t _n ∉U for every n ≥ 1. Evidently, any accumulation point of the sequence {t _n } belongs to $X_{\mathbb {I}}$.

It is immediate from the definition that $X_{\mathbb {I}}$ is compact and maps into itself by m _d . Of the d preimages of any point in $X_{\mathbb {I}}$, at least one lies outside U and therefore belongs to $X_{\mathbb {I}}$. This proves $m_d(X_{\mathbb {I}})=X_{\mathbb {I}}$. Finally, m _d restricted to $X_{\mathbb {I}}$ can be extended to a degree 1 monotone map $g: {\mathbb T} \to {\mathbb T}$ by setting g = m _d outside U and mapping each interval I _i to a point. □

Corollary 2.16

A non-empty compact m _d -invariant set X is a rotation set if and only if ${\mathbb T} \smallsetminus X$ contains d − 1 disjoint open intervals, each of length 1∕d.

Proof

Necessity follows from Theorem 2.7. For sufficiency, let $\mathbb {I}$ be the collection of the d − 1 disjoint intervals of length 1∕d in ${\mathbb T} \smallsetminus X$. By the above theorem $X_{\mathbb {I}}$ is a rotation set that contains X. Hence X itself is a rotation set. □

If Y  is a rotation set for m _d and if X ⊂ Y  is compact and m _d -invariant, then clearly X is also a rotation set for m _d , with ρ(X) = ρ(Y ). We record the following simple lemma for future reference:

Lemma 2.17

Suppose X, Y  are rotation sets for m _d containing more than one point, and assume X ⊂ Y . Then each major gap of X of multiplicity n contains n major gaps of Y  counting multiplicities.

Proof

Evidently each major gap of Y  is contained in a major gap of X. Let {I _i } be the collection of major gaps of X of multiplicities {n _i }. The number k _i of major gaps of Y  contained in I _i satisfies 0 ≤ k _i  ≤ n _i . Since $\sum k_i = \sum n_i =d-1$ by Theorem 2.7, we must have k _i  = n _i for all i. □

Let us call a rotation set maximal if it is not properly contained in another rotation set. Theorem 2.15 provides a convenient recipe for enlarging every rotation set to a maximal one.

Lemma 2.18

Every rotation set is contained in a maximal rotation set.

Proof

Suppose X is a rotation set for m _d . For each major gap (a, a + ℓ) of X of multiplicity n, consider the n disjoint subintervals (a + (j − 1)∕d, a + j∕d) for 1 ≤ j ≤ n. Let $\mathbb {I}$ denote the collection of the d − 1 disjoint open intervals of length 1∕d thus obtained. The rotation set $X_{\mathbb {I}}$ of Theorem 2.15 clearly contains X. Moreover, the endpoints of the intervals in $\mathbb {I}$ map to X under m _d , which shows they all belong to $X_{\mathbb {I}}$. Thus, $X_{\mathbb {I}}$ has d − 1 taut gaps of multiplicity 1. By Theorem 2.9 and Lemma 2.17, $X_{\mathbb {I}}$ is maximal. □

Corollary 2.19

A rotation set X for m _d is maximal if and only if it has d − 1 distinct gaps of length 1∕d. In this case $X=X_{\mathbb {I}}$ , where $\mathbb {I}$ is the collection of the major gaps of X.

The proof of Lemma 2.18 in fact gives the following improved lower bound for the number $N_{\max }(X)$ of the maximal rotation sets containing X:

Corollary 2.20

Suppose X is a rotation set for m _d with loose gaps I ₁, …, I _k of multiplicities n ₁, …, n _k . Then

$$\displaystyle \begin{aligned} N_{\max}(X) \geq \prod_{j=1}^k (n_j+1). \end{aligned}$$

In particular, X is contained in at least 2^k maximal rotation sets.

Proof

For each loose gap I = (a, a + ℓ) of X with multiplicity n, there are n + 1 different ways of choosing n disjoint subintervals of length 1∕d whose endpoints map to m _d (a) or m _d (a + ℓ) (the one in the proof of Lemma 2.18 was one of these choices). This leads to $\prod _{j=1}^k (n_j+1)$ different choices for the collection $\mathbb {I}$. □

Example 2.21

The 2-cycle $X= \{ \frac {1}{3} , \frac {2}{3} \}$ under doubling is contained in precisely two maximal rotation sets

$$\displaystyle \begin{aligned} X_{\mathbb{I}_1} = \Big\{ \frac{1}{3}, \frac{2}{3} \Big\} \cup \Big \{ \frac{1}{3}-\frac{1}{3 \cdot 2^{2n-1}} \Big\}_{n \geq 1} \cup \Big\{ \frac{2}{3} - \frac{1}{3 \cdot 2^{2n}} \Big\}_{n \geq 1} \end{aligned}$$

and

$$\displaystyle \begin{aligned} X_{\mathbb{I}_2} = \Big\{ \frac{1}{3}, \frac{2}{3} \Big\} \cup \Big \{ \frac{1}{3}+\frac{1}{3 \cdot 2^{2n}} \Big\}_{n \geq 1} \cup \Big\{ \frac{2}{3} + \frac{1}{3 \cdot 2^{2n-1}} \Big\}_{n \geq 1} \end{aligned}$$

corresponding to the collections $\mathbb {I}_1 = \{ (\frac {2}{3},\frac {1}{6}) \}$ and $\mathbb {I}_1 = \{ (\frac {5}{6},\frac {1}{3}) \}$. Note that each orbit in $X_{\mathbb {I}_i}$ eventually hits the 2-cycle X, and the intersection of $X_{\mathbb {I}_i}$ with the major gap of X is countably infinite.

The above example is a special case of a count for $N_{\max }(X)$ that we will establish in the next section for certain rational rotation sets (see Theorem 2.30). These rotation sets, however, are not typical. In fact, when d > 2 there are rational rotation sets for m _d that are contained in infinitely many maximal rotation sets. Here is an example:

Example 2.22

Consider the 2-cycle $X=\{ \frac {1}{4}, \frac {3}{4} \}$ under tripling. Define the sequences

$$\displaystyle \begin{aligned} t_n & = \sum_{j=0}^n \frac{1}{3^{2j+1}} + \frac{1}{3^{2n+1} \cdot 12} \\ s_n =m_3(t_n) & = \sum_{j=0}^n \frac{1}{3^{2j}} + \frac{1}{3^{2n} \cdot 12} \end{aligned} $$

for n ≥ 0. Then $\frac {1}{3} < t_0 < t_1 < t_2 < \cdots$ with $t_n \to \frac {3}{8}$ and $\frac {1}{12} = s_0 < s_1 < s_2 < \cdots$ with $s_n \to \frac {1}{8}$. For each n ≥ 0 the collection

$$\displaystyle \begin{aligned} \mathbb{I}_n = \Big\{ \Big( t_n ,t_n+\frac{1}{3} \Big) , \Big( \frac{3}{4},\frac{1}{12} \Big) \Big\} \end{aligned}$$

produces a rotation set $X_{\mathbb {I}_n}$ which evidently contains the 2-cycle X. The endpoints $\frac {3}{4}, \frac {1}{12}$ map to $\frac {1}{4}$ under m ₃, so they both belong to $X_{\mathbb {I}_n}$. The other endpoints $t_n ,t_n+\frac {1}{3}$ have the m ₃-orbit

$$\displaystyle \begin{aligned} t_n, \, t_n+\frac{1}{3} \mapsto s_n \mapsto t_{n-1} \mapsto s_{n-1} \mapsto \cdots \mapsto t_0 \mapsto s_0=\frac{1}{12} \mapsto \frac{1}{4} \end{aligned}$$

which, by monotonicity of {t _j } and {s _j }, never meets the pair of intervals in $\mathbb {I}_n$. This shows that both $t_n, t_n+\frac {1}{3}$ belong to $X_{\mathbb {I}_n}$. Thus $X_{\mathbb {I}_n}$ has a pair of major gaps of length $\frac {1}{3}$ and therefore is maximal by Corollary 2.19.

The situation in the irrational case is different and in fact simpler:

Theorem 2.23

Every irrational rotation set X for m _d is contained in finitely many maximal rotation sets. For any maximal rotation set Y ⊃ X and any gap I of X, the intersection Y ∩ I is finite (possibly empty) and eventually maps into X under the iterations of m _d .

Proof

Take any maximal rotation set Y ⊃ X. First suppose I is a major gap of X of multiplicity n. By Lemma 2.17 and Corollary 2.19, Y  has exactly n taut gaps of multiplicity 1 contained in I. We distinguish two cases:

• Case 1: I is taut. Then I has the form (a, a + n∕d) and

  $$\displaystyle \begin{aligned} Y \cap \bar{I} = \Big \{ a, a+\frac{1}{d}, \ldots, a+\frac{n}{d} \Big \}.\end{aligned} $$

  This condition uniquely determines the major gaps of Y  that are contained in I. Notice that the inclusion $m_d(Y \cap \bar {I}) \subset X$ holds.

• Case 2: I is loose. Consider the standard monotone map g _Y which is also an extension of m _d | _X . By Theorem 2.10, there is an i > 0 such that the interior J of $g_Y^{\circ i}(I)$ is a taut gap of X (there can be no periodic loose gap of X since ρ(X) is irrational). Note that $m_d^{\circ i}(Y \cap I)=g_Y^{\circ i}(Y \cap I)$ is contained in $Y \cap \bar {J}$ which is uniquely determined by Case 1. Hence the elements of Y ∩ I are among the finitely many $m_d^{\circ i}$-preimages of $Y \cap \bar {J}$. This gives finitely many choices for the major gaps of Y  in I.

The two cases above show that there are only finitely many choices for the major gaps of Y , hence for Y  itself by Theorem 2.9.

We have shown that for any major gap I of X, the intersection Y ∩ I is finite and eventually maps into X. Since every minor gap of X maps homeomorphically onto a major gap under some iterate of m _d , the result must also hold when I is minor. □

The number $N_{\max }(X)$ of maximal rotation sets Y  containing an irrational rotation set X depends on the structure of the gap graph Γ _X defined in the previous section. Suppose there is a maximal path in Γ _X of the form

$$\displaystyle \begin{aligned} I_k \to I_{k-1} \to \cdots \to I_1, \quad \text{with} \quad w(I_i)=n_i.\end{aligned} $$

(2.2)

Since I ₁ is taut, the major gaps of Y  in I ₁ are already determined. However, there are $n_1+n_2 \choose n_2$ choices for the major gaps of Y  in I ₂. For each of these choices, there are $n_1+n_2+n_3 \choose n_3$ choices for the major gaps of Y  in I ₃ and so on. This gives the count

$$\displaystyle \begin{aligned} N_{\max}(X) = \prod {\textstyle{n_1+n_2 \choose n_2} {n_1+n_2+n_3 \choose n_3} \cdots {n_1+\cdots+n_k \choose n_k}} = \prod \frac{(n_1+\cdots+n_k)!}{n_1! \, \cdots \, n_k!}, \end{aligned} $$

(2.3)

where the product is taken over all maximal paths in Γ _X of the form (2.2) (if there is no path in Γ _X , the product is taken over the empty set and is understood to be 1).

A quick inspection of Fig. 2.3 reveals that $N_{\max }(X)=1$ for d = 2, $N_{\max }(X) \leq 2$ for d = 3, and $N_{\max }(X) \leq 6$ for d = 4, and $N_{\max }(X) \leq 24$ for d = 5. More generally, we have the following

Theorem 2.24

$N_{\max }(X) \leq (d-1)!$ whenever X is an irrational rotation set for m _d .

Proof

If the gap graph Γ _X has no path, then $N_{\max }(X)=1$ and there is nothing to prove. Otherwise, let Γ _X have p ≥ 1 distinct maximal paths of the form (2.2), where the weights of the vertices in the i-th path add up to N _i , so N ₁ + ⋯ + N _p  ≤ d − 1. Then, by (2.3),

$$\displaystyle \begin{aligned} N_{\max}(X) \leq \prod_{i=1}^p N_i! \leq \Big( \sum_{i=1}^p N_i \Big)! \leq (d-1)! \end{aligned}$$

as required. □

2.3 Minimal Rotation Sets

A rotation set is called minimal if it does not properly contain another rotation set. This section will study the question of existence and uniqueness of minimal rotation sets that are contained in a given rotation set, in both rational and irrational cases.

Before we begin, a quick comment on topological dynamics is in order. The simple proof that minimality is equivalent to having all orbits dense requires a slight modification here, as the closure of an orbit in a rotation set is only forward invariant and may not be a rotation set.⁴ Similarly, the standard application of Zorn’s lemma to show that every rotation set contains a minimal rotation set needs some care because the intersection of a linearly ordered family of rotation sets is a priori forward invariant. This minor problem is addressed by observing that under m _d , every compact forward invariant set contains a compact invariant set. In fact, if Z is compact and satisfies m _d (Z) ⊂ Z, the nested intersection $K =\bigcap _{n \geq 0} m_d^{\circ n}(Z)$ is easily seen to satisfy m _d (K) = K.

Let us first consider the rational case, where minimal rotation sets are cycles. Let C = {t ₁, …, t _q } be a cycle of rotation number p∕q under m _d , where the t _j are in positive cyclic order and their subscripts are taken modulo q (see Sect. 1.3). By Theorem 1.14, $d t_j = t_{j+p}\, (\operatorname {mod} \ {\mathbb Z})$ for every j. The q gaps I _j  = (t _j , t _j+1) are permuted under any monotone extension g of m _d | _C , so $g(\bar {I}_j)=\bar {I}_{j+p}$. Recall that these gaps are either minor or loose: there can be no taut gap.

It follows from Theorem 2.3 that every rotation set X for m _d with ρ(X) = p∕q in lowest terms contains at least one q-cycle. But there could be several such minimal sets in X. For instance, under the tripling map m ₃, the union X = C ₁ ∪ C ₂ of the 3-cycles

$$\displaystyle \begin{aligned} C_1: \frac{4}{26} \mapsto \frac{12}{26} \mapsto \frac{10}{26} \quad \text{and} \quad C_2: \frac{7}{26} \mapsto \frac{21}{26} \mapsto \frac{11}{26} \end{aligned}$$

is a rotation set with $\rho (X)=\frac {2}{3}$. This can be seen, for example, from Corollary 2.16 since ${\mathbb T} \smallsetminus X$ contains the intervals $(\frac {12}{26}, \frac {12}{26}+\frac {1}{3})$ and $(\frac {21}{26},\frac {21}{26}+\frac {1}{3})$ on the circle. The general situation can be understood as follows.

We call a collection C ₁, …, C _N of distinct q-cycles under m _d with the same rotation number compatible if their union C ₁ ∪⋯ ∪ C _N is a rotation set. We say that C ₁, …, C _N are superlinked if for every pair i ≠ j, each gap of C _i meets C _j . Geometrically, this means that the points of C _i and C _j alternate as we go around the circle.

Lemma 2.25

C ₁, …, C _N are compatible if and only if they are superlinked.

In follows in particular that a collection of cycles are compatible if and only if they are pairwise compatible.

Proof

First suppose X = C ₁ ∪⋯ ∪ C _N is a rotation set. Consider the standard monotone map g = g _X , which is also a monotone extension of $m_d|{ }_{C_i}$ for each i. Pick any pair C _i , C _j . Since these cycles are distinct, there is a gap I of C _i that meets C _j at some point t. Then for every k ≥ 0, the interior J _k of g ^∘k(I) meets C _j at $g^{\circ k}(t)=m_d^{\circ k}(t)$. Since J ₀ = I, J ₁, …, J _q−1 form all the gaps of C _i , we conclude that C _i , C _j are superlinked.

Conversely, suppose C ₁, …, C _N are superlinked and consider the standard monotone map $g=g_{C_1}$. Take a gap I of C ₁ and let J be the interior of g(I). For 2 ≤ i ≤ N, let C _i  ∩ I = {a _i } and C _i  ∩ J = {b _i }. Using the fact that the C _i have the same rotation number, it is easy to see that b _i  = m _d (a _i ). As the C _i are superlinked, the points a _i appear in the same order in I as the points b _i in J, so there is an orientation-preserving homeomorphism h : I → J such that h(a _i ) = b _i for 2 ≤ i ≤ N. Repeating this process for every gap of C ₁ and gluing together the resulting homeomorphisms will then yield an orientation-preserving homeomorphism $h: {\mathbb T} \to {\mathbb T}$ which restricts to m _d on the union C ₁ ∪⋯ ∪ C _N . □

Theorem 2.26

The number of distinct cycles in a rational rotation set is bounded above by the number of its distinct major gaps.

In view of Theorem 2.7, we recover the following result of Goldberg as a special case (see [11] for the original combinatorial proof and [2] for an inductive argument reducing the problem down to d = 2):

Corollary 2.27

A rational rotation set for m _d contains at most d − 1 distinct cycles.

The upper bound d − 1 can always be achieved; see Corollary 3.15.

Proof of Theorem 2.26

Let Y  be a rational rotation set for m _d with ρ(Y ) = p∕q in lowest terms. Suppose C ₁, …, C _N are the distinct cycles in Y , all necessarily of length q. The union X = C ₁ ∪⋯ ∪ C _N is an m _d -invariant subset of Y , so it is a rotation set. By Lemma 2.25, the C _i are superlinked. It follows that any gap I of C ₁ contains precisely N gaps J ₁, …, J _N of X. Each J _i is periodic of period q and its orbit contains at least one major gap of X. Moreover, the orbits of J ₁, ⋯ , J _N are disjoint, so they cannot share any major gap of X. It follows that X, hence Y , has at least N distinct major gaps. □

Corollary 2.28

Every rational rotation set under the doubling map contains a unique cycle.

Example 2.29

Under the tripling map m ₃ there are five 4-cycles of rotation number $\frac {1}{4}$:

$$\displaystyle \begin{aligned} C_1: \ & \ \frac{1}{80} \, \mapsto \frac{3}{80} \, \mapsto \frac{9}{80} \, \mapsto \, \frac{27}{80} \vspace{2mm} \\ C_2: \ & \ \frac{2}{80} \, \mapsto \frac{6}{80} \, \mapsto \frac{18}{80} \, \mapsto \, \frac{54}{80} \vspace{2mm} \\ C_3=C_3+\frac{1}{2}: \ & \ \frac{5}{80} \, \mapsto \frac{15}{80} \, \mapsto \frac{45}{80} \, \mapsto \, \frac{55}{80} \vspace{2mm} \\ C_4=C_2+\frac{1}{2}: \ & \ \frac{14}{80} \, \mapsto \frac{42}{80} \, \mapsto \frac{46}{80} \, \mapsto \, \frac{58}{80} \vspace{2mm} \\ C_5=C_1+\frac{1}{2}: \ & \ \frac{41}{80} \, \mapsto \frac{43}{80} \, \mapsto \frac{49}{80} \, \mapsto \, \frac{67}{80} \end{aligned} $$

By Corollary 2.27, at most two 4-cycles under tripling can be compatible. By Lemma 2.25, this happens precisely when the two 4-cycles are superlinked. Simple inspection shows that (C ₁, C ₂), (C ₂, C ₃), (C ₃, C ₄) and (C ₄, C ₅) are the only compatible pairs (compare Fig. 2.4).

Fig. 2.4

The five 4-cycles of rotation number $\frac {1}{4}$ under tripling, shown in different colors (angles are given in multiples of $\frac {1}{80}$). Only the four superlinked pairs (red, blue), (blue, green), (green, yellow), and (yellow, brown) are compatible cycles

Before moving on to the irrational case, let us use the above ideas to show that for some rational rotation sets the lower bound of Corollary 2.20 is sharp:

Theorem 2.30

Let X be a rational rotation set for m _d which is the union of d − 1 distinct cycles. Then $N_{\max }(X)=2^{d-1}$.

Proof

By Theorem 2.26 X has d − 1 major gaps, all loose and of multiplicity 1. If ρ(X) = p∕q, these major gaps have disjoint orbits which are periodic of period q. Let Y  be any maximal rotation set containing X. Each major gap I = (a, b) of X contains a single major gap J of Y  of length 1∕d. We claim that J = (a, a + 1∕d) or J = (b − 1∕d, b). Otherwise J = (t, t + 1∕d), where a < t < t + 1∕d < b. The standard monotone map g = g _Y is also a monotone extension of m _d | _X , so g ^∘q maps I onto itself fixing the endpoints a, b. Moreover, the gaps g(I), …, g ^∘q−1(I) of X are all minor, so they cannot contain major gaps of Y ; as such, g acts homeomorphically on them. It follows that g ^∘q is homeomorphic on [a, t] ∪ [t + 1∕d, b] and collapses J to the single point $m_d^{\circ q}(t)$. This image point necessarily lies in J since $g^{\circ q}=m_d^{\circ q}$ is expanding on both [a, t] and [t + 1∕d, b]. This is a contradiction since $m_d^{\circ q}(t) \in Y$.

Thus, there are just two possibilities for each major gap of Y  inside a given major gap of X, hence 2^d−1 possibilities altogether for the major gaps of Y , and therefore for Y  itself. This proves $N_{\max }(X) \leq 2^{d-1}$. The result now follows since $N_{\max }(X) \geq 2^{d-1}$ by Corollary 2.20. □

The following corollary immediately follows from the above theorem and its proof:

Corollary 2.31

Every rotation cycle X under the doubling map is contained in exactly two maximal rotation sets. Moreover, if (a, b) is the major gap of X, then the intervals $(a,a+\frac {1}{2})$ and $(b-\frac {1}{2},b)$ are the major gaps of these maximal rotation sets.

Compare Example 2.21.

Example 2.32

Consider the 2-cycle $X=\{ \frac {1}{4}, \frac {3}{4} \}$ under tripling. We showed in Example 2.22 that $N_{\max }(X)=\infty$. However, the enlarged rotation set $Y=\{ \frac {1}{8}, \frac {1}{4}, \frac {3}{8}, \frac {3}{4} \}$, a union of two 2-cycles under tripling, has $N_{\max }(Y)=4$ by Theorem 2.30!

We now consider minimal rotation sets in the irrational case.

Theorem 2.33

Every irrational rotation set X for m _d contains a unique minimal rotation set K. Moreover,

• (i) K is the Cantor attractor of any monotone extension of m _d | _X .

• (ii) Each gap of K contains at most finitely many points of X, all of which eventually map to K under the iterations of m _d .

Proof

Take a monotone extension g of m _d | _X and let K be the Cantor attractor of g, as in Theorem 1.20. Let Z be any non-empty compact m _d -invariant subset of X. By Theorem 1.20, K = ω _g (t) ⊂ Z for every t ∈ Z. It follows that K is the unique minimal rotation set contained in X.

To verify the second statement, let Y  be any maximal rotation set containing X (whose existence is guaranteed by Lemma 2.18). Since Y  contains K, Theorem 2.23 shows that for each gap I of K, the intersection Y ∩ I is at most finite and maps into K under the iterations of m _d . Hence the same must be true of X ∩ I. □

By (the proof of) Theorem 1.20, the gaps of the Cantor attractor of g are the plateaus of the Poincaré semiconjugacy φ between g and r _θ . Thus, we have the following

Corollary 2.34

Suppose X is a minimal rotation set for m _d with ρ(X) = θ irrational. Then there exists a degree 1 monotone map $\varphi : {\mathbb T} \to {\mathbb T}$ , whose plateaus are precisely the gaps of X, which satisfies φ ∘ m _d  = r _θ  ∘ φ on X.

Here is the converse statement. Recall that for each point $s \in {\mathbb T}$, I _s denotes the interior of the fiber E _s  = φ ⁻¹(s).

Theorem 2.35

Let θ be irrational and $\varphi : {\mathbb T} \to {\mathbb T}$ be a degree 1 monotone map with the property that I _s  ≠ ∅ implies I _s−θ ≠ ∅. Denote by X the complement of the union of all plateaus of φ. If

$$\displaystyle \begin{aligned} \varphi \circ m_d = r_{\theta} \circ \varphi \quad \mathit{\text{on}} \quad X, \end{aligned} $$

(2.4)

then X is a minimal rotation set for m _d with ρ(X) = θ.

Proof

The assumptions imply that φ has plateaus; otherwise $X={\mathbb T}$ and (2.4) would exhibit a global conjugacy between the degree d ≥ 2 map m _d and the rotation r _θ , which is impossible.

We invoke Theorem 1.22 to find a degree 1 monotone map $g: {\mathbb T} \to {\mathbb T}$ such that φ ∘ g = r _θ  ∘ φ on ${\mathbb T}$. Then X is the Cantor attractor of g. If t ∈ X is not an endpoint of a plateau and s = φ(t), then E _s  = {t}, so by the assumption E _s+θ is a singleton {t′}. The semiconjugacy relation (2.4) for m _d and the one for g then show that m _d (t) = t′ = g(t). Since the set of such t is dense in X, we conclude that g = m _d on X. As the Cantor attractor of g, X is minimal for g and hence for m _d , and m _d (X) = g(X) = X. This completes the proof that X is a minimal rotation set. □

We conclude this section with characterizations of minimal rotation sets, as well as those that are both minimal and maximal.

Theorem 2.36

A rotation set for m _d is a Cantor set if and only if it is minimal and has irrational rotation number.

Proof

The “if” part follows from Theorem 2.33. For the “only if” part, suppose X is a Cantor set. Then ρ(X) is irrational since a rational rotation set is at most countable (Theorem 2.3). Let K be the unique minimal rotation set contained in X. If K ≠ X, some gap I of K would have to meet X. But then by Theorem 2.33 the intersection X ∩ I would be finite, consisting of isolated points of X. This would contradict the assumption that X is a Cantor set. □

Let us call a rotation set exact if it is both minimal and maximal.⁵ Evidently a rational rotation set can never be exact. In the irrational case, the following criterion follows immediately from Corollary 2.19 and Theorem 2.36:

Theorem 2.37

An irrational rotation set for m _d is exact if and only if it is a Cantor set with d − 1 distinct gaps of length 1∕d.

Corollary 2.38

Every irrational rotation set under the doubling map is exact.

Proof

Let X be an irrational rotation set under doubling. Then X has a single major gap I of multiplicity 1 which is necessarily taut by Corollary 2.11. If K is the unique minimal rotation set contained in X, then K is a Cantor set with a single taut gap of multiplicity 1 which can only be I. It follows from Theorem 2.9 that K = X, and then from Theorem 2.37 that X is exact. □

Remark 2.39

The above corollary is false in higher degrees. For example, there are minimal irrational rotation sets under tripling with a pair of major gaps of lengths $\frac {1}{3}$ and $\frac {4}{9}$ which therefore are not maximal (compare Theorem 4.31). However, every irrational rotation set under tripling is either minimal, or maximal, or both. In every degree > 3, there are irrational rotation sets that are neither minimal nor maximal.

For more on the role of exact rotation sets, see Sect. 4.3.