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Weak Disks of Denjoy Minimal Sets

arXiv:math/9209220v1 [math.DS] 1 Sep 1992Weak Disks of Denjoy Minimal SetsPhilip BoylandInstitute for Mathematical SciencesSUNY at Stony BrookStony Brook, NY 11794Internet: boyland@math.sunysb.eduAbstract. This paper investigates the existence of Denjoy minimal sets and,more generally, strictly ergodic sets in the dynamics of iterated homeomorphisms.It is shown that for the full two-shift, the collection of such invariant sets with theweak topology contains topological balls of all finite dimensions.

One implicationis an analogous result that holds for diffeomorphisms with transverse homoclinicpoints. It is also shown that the union of Denjoy minimal sets is dense in thetwo-shift and that the set of unique probability measures supported on these setsis weakly dense in the set of all shift-invariant, Borel probability measures.Section 0: Introduction.

One strategy for understanding a dynamical system isto first isolate invariant sets that are dynamically indecomposable. One then studies thestructure of these pieces and how they fit together to give the global dynamics.

This ideagoes back at least to Birkhoffand has a particularly clear expression in Conley’s Morsedecompositions.There are many notions of dynamical indecomposibility in the literature.In thispaper we consider a fairly strong one that uses both topology and measure. An invariantset is called strictly ergodic if it is both minimal (every orbit is dense) and uniquely ergodic(existence of a unique, invariant Borel probability measure).

These properties are preservedunder topological conjugacy but not measure isomorphism.The simplest such invariant sets are periodic orbits, and there are many theoremsconcerning their existence. The next simplest strictly ergodic systems are probably rigidrotations on the circle with irrational rotation number and the closely related Denjoyminimal sets.

Elements of these invariant sets are sometimes called (generalized) quasi-periodic points. The models for Denjoy minimal sets are the minimal sets in nontransitivecircle homeomorphisms with irrational rotation number.

An abstract dynamical system iscalled a Denjoy minimal set if it is topologically conjugate to such a model. One of thequestions that motivated this paper is what kind of properties of periodic orbits are alsotrue for more general strictly ergodic invariant sets, in particular, for Denjoy minimal sets?One way to begin to address this question is to collect these invariant sets into spaces.For a fixed homeomorphism f of a compact metric space X, let S(X, f) denote the setof all strictly ergodic f-invariant subsets of X.

Since different minimal sets are of neces-sity disjoint, each point in S(X, f) represents a minimal set that is disjoint from everyother minimal set. A strictly ergodic set supports a unique invariant Borel probabilitymeasure, so we may use these measures with the weak topology to put a topology onS(X, f).

If D(X, f) denotes the set of f-invariant subsets that are Denjoy minimal sets,then D(X, f) ⊂S(X, f), so we may use the weak topology on D(X, f) also.1

In ([M]), Mather shows that for a area-preserving monotone twist map of the annulus,f : A →A, the nonexistence of an invariant circle with a given irrational rotation numberimplies the existence of numerous Denjoy minimal sets with that rotation number. Moreprecisely, using the notation just introduced, D(A, f) contains topological balls of everyfinite dimension.From one point of view this is a very surprising result.One has anarbitrarily large dimensional family of minimal sets embedded in a two-dimensional dy-namical system.

Another question that motivated this paper is how common is this kindof phenomenon in dynamics on finite dimensional manifolds?It is important to note that even for a smooth system, S(M, f) can be empty. Oneexample of this is Furstenberg’s Cω-diffeomorphism of the two torus that is minimal butnot strictly ergodic ([F]).

However, for the full shift on two symbols (Σ2, σ) one has:Theorem 0.1. The space S(Σ2, σ) contains a subspace homeomorphic to the Hilbertcube and the space D(Σ2, σ) contains topological balls of dimension n for all natural num-bers n.The basic tool in the proof of this theorem is the main construction.

This construc-tion takes a certain type of open set in the circle (a regular one) and produces a compact,invariant set in the full two-shift. The construction uses the open set to produce itinerarieswith respect to a rigid rotation on the circle by an irrational angle.

This process is some-what analogous to using a Markov partition to produce a symbolic model for a system.Another analogous process is used in the kneading theory of unimodal maps of the interval.The difference here is that the chosen open set, in general, has no relation to the dynam-ics. The Hilbert cube of strictly ergodic sets is obtained by showing that the invariantsets constructed in the two-shift have unique invariant probability measures that dependcontinuously on the regular open sets in the appropriate topologies.The main construction is a generalization of Morse and Hedlund’s construction ofSturmian minimal sets as described on page 111 of [G-H].

Such generalizations are astandard tool in topological dynamics. In particular, the main construction is a specialcase of the almost automorphic minimal extensions of Markley and Paul given in [M-P].Also of particular relevance are pages 234-241 of [A] and [H-H1].For any regular open set, the main construction yields a minimal set in the shift.

Ifthe open set is a finite union of intervals, it gives a Denjoy minimal set. When the openset is more complicated, the resulting minimal set is more complicated.

In particular, itfollows from [M-P] that for certain open sets the construction gives minimal sets that havepositive topological entropy and are not uniquely ergodic (see Remark 3.4 below).The full two-shift is frequently embedded in the iterates of a complicated dynamicalsystem. (In fact, this is one definition of a “complicated” dynamical system.) In view ofTheorem 0.1 one would therefore expect that that S(X, f) will frequently contain a Hilbertcube.

In the following corollary, the first sentence is a consequence of Theorem 0.1 and theBirkhoff-Smale theorem (a particularly suitable statement of which can be found on page109 of [Rl]). The second sentence follows from the first and a theorem of Katok ([K]).Corollary 0.2.

If f : M →M is a diffeomorphism of the compact manifold M thathas a transverse homoclinic orbit to a hyperbolic periodic point, then S(M, f) containsa subspace homeomorphic to the Hilbert cube. In particular, this is the case when M is2

two-dimensional, f is C1+α and has positive topological entropy.As was the case with Mather’s theorem, one has a large dimensional family of minimalsets (in this case an infinite dimensional family) embedded in finite dimensional dynamics.We shall see in Remark 3.7 below that in many cases this can be viewed as a manifestationof the fact that the Hilbert cube is the continuous, surjective image of the Cantor set.There is an invariant Cantor set ˆΛ embedded in the dynamics. The orbit closure of eachpoint in ˆΛ supports a unique invariant probability measure.

When the measures are giventhe weak topology, the map that takes the point to the measure is a continuous surjectionof the Cantor set ˆΛ onto the Hilbert cube.There are two important examples that illustrate the necessity of the smoothness anddimension in the second sentence of Corollary 0.2. In [R] Rees constructs a homeomorphismof the two torus that is minimal and has positive topological entropy.

Herman gives aCω-difeomorphism of a 4-manifold that is also minimal with positive topological entropy([Hm]). Neither example is uniquely ergodic, so in these cases S(M, f) is empty.

Thesecond sentence of Corollary 0.2 also raises the question of a converse.Specifically, ifS(M, f) contains a subspace that is homeomorphic to the Hilbert cube, does f have positivetopological entropy? Proposition 3.1 shows that this is false on manifolds of dimensionbigger than three.It is an easy exercise to show that periodic orbits are dense in the full two-shift.

Asomewhat deeper result due to Parthasarathy says that the invariant probability measuressupported on period orbits are weakly dense in the set of all shift-invariant probabilitymeasures, M(Σ2, σ) ([P]). The next proposition gives the analog of these results for Denjoyminimal sets.Proposition 0.3.

(a) The set of points that are members of Denjoy minimal sets is dense in Σ2. (b) The set of invariant measures supported on Denjoy minimal sets is weakly densein the set of invariant measures, i.e.

D(Σ2, σ) is dense in M(Σ2, σ).This paper is organized as follows.Section 1 gives basic definitions, backgroundinformation and the main construction. Section 2 contains the statement and proof ofthe main theorem.

This theorem describes continuity properties of the main constructionand the structure of resulting invariant sets. Section 2 also contains the proof of Theorem0.3.

The proof of Theorem 0.1 is given in Section 3, as is the example that shows thatthe converse of Corollary 0.2 is false in dimensions three and greater. The last sectionexamines the relationship between the intrinsic rotation number of a Denjoy minimal setand its “extrinsic” rotation number when it is embedded in a map of the annulus.

It isalso shown that any Denjoy minimal set in the two-shift can be generated from a regularopen set in the circle using the main construction.Acknowledgments: The author would like to thank B. Kitchens, N. Markley, B.Weiss and S. Williams for useful comments and references.Section 1: Preliminaries. This section introduces assorted notation and definitionsand recalls some basic facts from topology, ergodic theory and topological dynamics.

Many3

of these facts are stated without proof or references. In such cases, the facts are eitherelementary exercises or can be found in Walters’ book [W].For a set X, Cl(X), Int(X), Xc and Fr(X) denote the closure, interior, complementand frontier of the set, respectively.

The operator ⊔is the disjoint union. Thus A ⊔Brepresents the union of the two sets, but conveys the added information that the sets aredisjoint.

The indicator function of a set X is denoted IX. Thus IX(x) = 1 if x ∈X,and is 0 otherwise.

The circle is S1 = R/Z and Rη : S1 →S1 is rigid rotation by η, i.eRη(θ) = θ + η mod 1. Haar measure on the circle is denoted by m.A nonempty, proper subset U ⊂S1 is called a regular open set if Int(Cl(U)) = U.The set of all regular open sets isRO = {U ⊂S1 : U is a regular open set}.Given an open set U, its ∗-dual is the interior of its complement and is denoted by U ∗=Int(U c).

Note that U is regular open if and only if S1 can be written as the disjoint unionof three nonempty sets, S1 = U ⊔F ⊔U ∗with F = Fr(U) = Fr(U ∗). In consequence,U ∈RO if and only if U ∗∈RO.The set RO of regular open sets will be topologized using the symmetric difference ofsets.

For U, V ∈RO, their symmetric difference is U△V = (U ∩V c) ⊔(U c ∩V ) and thedistance between them is d(U, V ) = m(U△V ). If U and V are regular open, when U△Vis nonempty it contains an interval.

In particular, d(U, V ) = 0 if and only if U = V . Sinced(U, V ) =R|IU −IV |dm = ∥IU −IV ∥1, RO maybe thought of as a subspace of L1(S1, m).This makes it clear that d gives a metric on RO.If the frontiers of either U or V have positive measure, it could happen that d(U, V ) ̸=d(U ∗, V ∗).

To avoid this and related situations it is sometimes necessary to restrict atten-tion to the set of regular open sets whose frontiers have measure zero,RO0 = {U ∈RO : m(Fr(U)) = 0}.A metric that controls both regular open sets and their ∗-duals is given byd∗(U, V ) = (d(U, V ) + d(U ∗, V ∗))/2.Unless otherwise noted, the topology on RO will be that given by the metric d∗. Notethat when restricted to RO0, d and d∗give the same metric.It will also be useful to identify regular open sets that are equal after a rigid rotationof S1.

More precisely, say U ∼V if there exists an η ∈S1 with V = Rη(U). Denotethe quotient spaces by RO′ = RO / ∼and RO′0 = RO0 / ∼.

Note that the topologygenerated by the projection RO →RO′ can be viewed as being generated by the metricd′([U], [V ]) = inf{d∗(U, Rη(V )) : η ∈S1}, where [U] denotes the equivalence class of Uunder ∼.A related notion is that of a symmetric set. A set U ∈RO is called symmetric if thereexists an η ̸= 0 with Rη(U) = U.

Because U is open, such an η will always be a rationalnumber.In this paper a dynamical system means a pair (X, h) where X is a compact met-ric space and h is a homeomorphism.Given a point x ∈X, its orbit is o(x, h) =4

{. .

. , h−1(x), x, h(x), .

. .

}.A finite piece of the forward orbit is denoted o(x, h, N) ={x, h(x), . .

., hN(x)}.If (X, h) →(Y, g) is a continuous semiconjugacy, then (X, h) iscalled an extension of (Y, g), and (Y, g) is a factor of (X, h). When the semiconjugacy isone to one on a dense Gδ set, the extension is termed almost one to one.The pair (X, h) is called a minimal set if every orbit is dense.

The pair is uniquely er-godic if there exists a unique invariant Borel probability measure. A useful characterizationis: (X, h) is uniquely ergodic if and only if the sequence of functions (PNi=0 f ◦hi)/(N +1)converges uniformly for all f ∈C(X, R).

A pair that is both minimal and uniquely er-godic is called strictly ergodic. Note that the property of being minimal, uniquely ergodicor strictly ergodic is preserved under topological conjugacy.

Also, if an extension is strictlyergodic, then so is its factor.A compact h-invariant set Y ⊂X is called minimal, uniquely ergodic or strictlyergodic if h restricted to Y has that property. In a slight abuse of notation, this situationis described by saying that (Y, h) is minimal, etc.Perhaps the simplest nontrivial strictly ergodic system is (S1, Rα) for an irrational α.A homeomorphism g : S1 →S1 that has an irrational rotation number and the pair (S1, g)is not minimal is called a Denjoy example.

Such examples are classified up to topologicalconjugacy in [My]. The two classifying invariants are the rotation number and the set oforbits that are “blown up” into intervals.

A Denjoy example always has a unique minimalset Y ⊂S1 with (Y, g) strictly ergodic.An abstract dynamical system (X, h) is called a Denjoy minimal set if it is topologi-cally conjugate to the minimal set in a Denjoy example. Such an (X, h) is always strictlyergodic.

Mather points out in [M] that a Denjoy minimal set (X, h) always has a welldefined intrinsic rotation number, i.e. if (X, h) is topologically conjugate to the minimalsets in two Denjoy examples (S1, g1) and (S1, g2), then either g1 and g2 have the samerotation number or else g1 and g−12do.

If (X, h) is a Denjoy minimal set with intrinsicrotation number α, it is an almost one to one extension of (S1, Rα).A general dynamical system (Z, h) can have many invariant subsets that are Denjoyminimal sets or strictly ergodic. These subsets are collected together in the spacesD(Z, h) = {Y ⊂Z : (Y, h) is a Denjoy minimal set}andS(Z, h) = {Y ⊂Z : (Y, h) is strictly ergodic}.To topologize these spaces we recall the weak topology on measures.

Given a dynami-cal system (Z, h), the set of all its invariant, Borel probability measures is denoted M(Z, h).The weak topology on M can be defined by saying that the measures µn →µ0 weakly ifand only ifRfdµn →Rfµ0 for all f ∈C(Z, R). Note that M(Z, h) with this topology iscompact, and when viewed as a subspace of the dual space to C(Z, R), it is convex withextreme points equal to the ergodic measures.

Since a strictly ergodic system supports aunique invariant probability measure, there is a natural inclusion S(Z, h) ⊂M(Z, h). Thisinclusion induces a topology on S(Z, h) that will be called the weak topology.

The factthat D(Z, h) ⊂S(Z, h) allows us to use the weak topology on D(Z, h) also.In the absence of unique invariant measures we use the Hausdorffmetric to measurethe distance between compact invariant subsets.Given a compact space X, the space5

consisting of the closed subsets of X with the Hausdorfftopology is denoted H(X). Notethat if X is compact metric then so is H(X).

A map Φ : E →H(X) is called lowersemicontinuous if for all closed subsets Y ⊂X, the set {e ∈E : Φ(e) ⊂Y } is closed in E.We will need the fact that the following property implies that Φ is lower semicontinuous:When en →e in E and for some subsequence {ni}, Φ(eni) →K in H(X) then Φ(e) ⊂K.Informally, Φ is lower semicontinuous if when you perturb e, Φ(e) may get suddenly larger,but never suddenly smaller.The full shift on two symbols is the the pair (Σ2, σ) consisting of the sequence spaceΣ2 = {0, 1}Z and the shift map σ. A symbol block b is a finite sequence b0, b1, .

. ., bN−1which each bi equal to 0 or 1.

The length of the block b is N and the period is its periodwhen considered as a cyclic word. A sequence s ∈Σ2 has initial block b if bi = si fori = 0, .

. ., N −1.It is notationally convenient to view the topology on Σ2 as beinggenerated by a metric dΣ with dΣ(s, t) < 1/N if and only if si = ti for |i| < N. A cylinderset depends on a block b and an integer n and is a set of the formCnb = {s ∈Σ2 : si+n = bi, for i = 0, .

. ., length(b) −1}.If n = 0, we write C0b = Cb.Since cylinder sets are both open and closed, their indicator functions are continuous.In fact, the finite linear combinations of such indicator functions form a dense set inC(Σ2, R).

This implies that the measures µn →µ0 weakly if and only if µn(Cnb ) →µ0(Cnb )for all cylinder sets Cnb . Since the elements of M(Σ2, σ) are shift invariant measures, anysuch measure µ satisfies µ(Cnb ) = µ(Cb) for all n. Thus the topology on M(Σ2, σ) is infact generated by the metricd(µ1, µ2) =X|µ1(Cb(n)) −µ2(Cb(n))|/2nwhere the sum is over some enumeration b(n) of all possible blocks by the natural numbersn.The main construction in this paper takes a regular open set in the circle and producesa compact invariant set in (Σ2, σ) along with an invariant measure.As noted in theintroduction, it is closely related to the construction given in [M-P].

We are primarilyinterested here in the dependence of the construction on the open set and a “rotationnumber”. This dependence is encoded in two functions λ : RO0 × S1 →M(Σ2, σ) andΛ : RO × S1 →H(Σ2) defined as follows.Fix U ∈RO and r ∈S1 Define B ⊂S1 asB = {x ∈S1 : o(x, Rr) ∩Fr(U) = ∅}.Since Uis regular open,Fr(U) is closed and nowhere dense,and thus sinceB = ∩i∈NRir(Fr(U)c), B is dense Gδ.

Now define φ : B →Σ2 so that(φ(x))i = IU(Rir(x)).Thus for any point x ∈B, the sequence φ(x) is the “itinerary” of x under Rr with respectto the set U, i.e. φ(x) has a 1 in the ith place if Rir(x) is in U and 0 if it is in U ∗.

It iseasy to see that φ is continuous.6

Now define Λ(U, r) = Cl(φ(B)). If U ∈RO0, then m(U ⊔U ∗) = 1 and so m(B) = 1.Thus we may define a probability measure λ ∈M(Σ2, σ) by λ = φ∗(m), where as usualthis means that λ(X) = m(φ−1(X)) for a Borel set X.In this construction, B and φ depend on the choice of U and r. If this dependenceneeds to be emphasized, we write B = BU,r and φ = φU,r.

It is clear that for all η ∈S1and U ∈RO, Λ(Rη(U), r) = Λ(U, r) and λ(Rη(U), r) = λ(U, r). Thus the maps Λ and λdescend to maps on RO′ × S1 and RO′0 × S1 that will also be called Λ and λ.To make the last definition, we need to adopt the notation that U0 = U ∗and U1 = U.For a block of symbols b of length N + 1, defineUb,r =N\i=0R−ir (Ubi).The important property of these sets is that for x ∈B, x ∈Ub,r if and only if φ(x) is inthe cylinder set Cb.

As a consequence, for U ∈RO0, λ(U, r)[Cb] = φ∗m(Cb) = m(Ub,r).Lemma 1. The following maps are continuous.

(a) For fixed U ∈RO, the map S1 →R given by η 7→d∗(U, Rη(U)). (b) For fixed U ∈RO, the map S1 →RO given by η 7→Rη(U).

(c) The map RO →R given by U 7→m(U). (d) For fixed symbol block b, the map RO × S1 →RO given by (U, r) 7→Ub,r.Proof of (a) and (b).

We first prove continuity of the map η 7→d(U, Rη(U)) atη = 0. Since U ∈S1 is open, we can find a countable set of disjoint intervals {In} so thatU = ⊔In.

Now given ǫ > 0, pick M so that Pn>M m(In) < ǫ/4 and assume |η| < ǫ/(4M).Now for each n, clearly m(In ∩Rη(U)c) < η and som(U ∩Rη(U)c) Mm(In) +Xn≤Mm(In ∩Rη(U)c)< ǫ/2.Now since m(U c ∩Rη(U)) = m(R−η(U)c ∩U), we also get m(U c ∩Rη(U)) < ǫ/2 and sod(U, Rη(U)) < ǫ.What we have just shown also implies that η 7→d(U ∗, Rη(U ∗)) is continuous atη = 0, and thus η 7→d∗(U, Rη(U)) is also. Since d(Rη(U), Rη′(U)) = d(U, Rη−η′(U)),the continuity of η 7→Rη(U) at all η follows.

Finally, since d∗is a metric, and therefore acontinuous function RO × RO →R, we get η 7→d∗(U, Rη(U)) continuous for all η.Proof of (c). Given two finite collections of sets Ai and Bi with i ∈{0, .

. .N} usingthe fact that d(A, B) = ∥IA −IB∥1 and standard integral inequalities it is easy to showthat |m(A) −m(B)| ≤d(A, B) and d∗(∩Ai, ∩Bi) ≤P d∗(Ai, Bi).The continuity of U 7→m(U), follows from the fact that d∗(U, V ) ≤ǫ/2 impliesǫ ≥d(U, V ) ≥|m(U) −m(V )|.Proof of (d).

If the length of the fixed block b is N + 1, then given ǫ > 0 using (a),pick δ < ǫ/(2N + 2) so that |η| < δ implies d∗(U, Rη(U)) < ǫ/(2N + 2).7

We therefore have for (V, s) ∈RO × S1 with d(U, V ) < δ and |r −s| < δ/N,d∗(Ub,r, Vb,s) = d∗(∩R−ir (Ubi), ∩R−is (Vbi))≤Xd∗(R−ir (Ubi), R−is (Vbi))=Xd∗(Ri(s−r)(Ubi), Vbi)≤X(d∗(Ri(s−r)(Ubi), Ubi) + d∗(Ubi, Vbi)≤ǫ.⊔⊓Section 2: The main theorem.The main goal of this section is to prove thefollowing theorem. For the reader interested in the quickest route to Theorem 0.1, we notethat the lower semicontinuity of Λ and the results in part (3) are not needed for that proof.Theorem 2.

Let the maps λ : RO′0 × S1 →M(Σ2, σ) and Λ : RO′ × S1 →H(Σ2)be as defined in Section 1. (1) The map λ is continuous and the map Λ is lower semicontinuous.

(2) Fix α ̸∈Q. (a) For all U ∈RO′, (Λ(U, α), σ) is an almost one to one minimal extensionof (S1, Rnα) for some natural number n.(b) If U ∈RO′0, then (Λ(U, α), σ) is uniquely ergodic.

(c) If Fr(U) is a finite set, then (Λ(U, α), σ) is a Denjoy minimal set withintrinsic rotation number nα for some natural number n.(d) For fixed α ̸∈Q, when considered as a function of U, Λ and λ areinjective. (3) Fix p/q ∈Q with p and q relatively prime.

(a) For all U ∈RO′, Λ(U, p/q) is a finite collection of periodic orbits whoseperiods divide q. (b) For fixed p/q ∈Q, when considered as a function of U, the image of λis the convex hull of the probability measures supported on the periodicorbits whose periods divide q.Proof of (1).

Since Λ(Rη(U), r) = Λ(U, r), it suffices to check the continuity of Λ asa map defined on RO. A similar comment holds for λ.As noted in the previous section, the weak topology on M(Σ2, σ) is generated by themetric d(λ1, λ2) = P |λ1(Cb(n)) −λ2(Cb(n))|/2n and λ(U, r)[Cb] = m(Ub,r).

Thus to provethe continuity of λ it suffices to check that for fixed b the map U 7→m(Ub,r) is continuous.This follows from Lemma 1 (c) and (d).For the proof of the lower semicontinuity of Λ, begin by assuming that (U (n), r(n)) →(U (0), r(0)). If for some subsequence {ni}, Λ(U (ni), r(ni)) →K in the Hausdorfftopology,then we will show that Λ(U (0), r(0)) ⊂K.

As noted in the previous section, this impliesthe desired semicontinuity. Fix an x0 ∈B(0) and integer N > 0 and let b be the initialblock of length N + 1 in φ(x0).

This certainly implies that U (0)b,r(0) is a nonempty openset and therefore has positive measure. Therefore by Lemma 1 (c) and (d) there exists8

an M so that n > M implies that m(U (n)b,r(n)) > 0. In particular, for n > M, there existsxn ∈B(n) so that φn(xn) has its initial block equal to b.

There therefore exits a sequencexj ∈B(j) with φj(xj) →φ0(x0).Now assuming that for some subsequence {ni}, Λ(U (ni), r(ni)) →K in the Hausdorfftopology, then if xni ∈B(ni) is the appropriate subsequence of the sequence constructed inthe previous paragraph, then φni(xni) →φ0(x0), so certainly φ0(x0) ∈K. But x0 ∈B(0)was arbitrary, and so φ0(B(0)) ⊂K and since Λ(U (0), r(0)) is the closure of the φ0(B(0)),we have Λ(U (0), r(0)) ⊂K, as required.Proof of (2).

For the proof of (2), fix an α ̸∈Q and for the proof of (2a), (2b) and(2c) a U ∈RO. We will suppress the dependence of various objects on U and α and soΛ = Λ(U, α), etc.(2a).

To prove the minimality of Λ we use the following characterization of minimality([O]): If f : X →X is a homeomorphism of a compact metric space and x ∈X, thenCl(o(x, f)) is a minimal set if and only if given ǫ > 0, there exists an N such that for alln, there exists an i with 0 ≤i ≤N and d(f n+i(x), x) < ǫ.To apply this to the case at hand, first note that for x ∈B, certainly o(x, Rα) is densein B, and so Λ = Cl(o(φ(x), σ)). Since (S1, Rα) is minimal, the above property holds forCl(o(x, Rα)).

Since φ is continuous, it also holds for Cl(o(φ(x), σ)) = Λ, which is thereforeminimal.The proof of the semiconjugacy requires a new definition. Given U, V ∈RO, defineρ(U, V ) = sup{m(I) : I is an interval contained in U△V }.Now ρ will not satisfy thetriangle inequality but it is easy to see that for fixed U ∈RO, the map η 7→ρ(U, Rη(U))is a continuous function S1 →R.

Also, if U is asymmetric, then ρ(U, Rη(U)) = 0 if andonly if η = 0.The first step in the proof of the semiconjugacy is to show that φ is injective whenU is asymmetric. Assume that for x1, x2 ∈B, φ(x0) = φ(x1), and therefore for all i,IU(Riα(x1)) = IU(Riα(x2)).

Thus if x2 = Rη(x1), IU = IU ◦Rη when restricted to the denseset o(x1, Rα). In particular, ρ(U, Rη(U)) = 0 and since U is asymmetric, d(x1, x2) = η = 0.Continuing with the assumption that U is asymmetric, we show that φ−1 is uniformlycontinuous.

Since φ(B) is certainly dense in Λ, this implies that we can extend φ−1 to asemiconjugacy from (Λ, σ) to (S1, Rα).Since S1 is compact and η 7→ρ(U, Rη(U)) is continuous, given ǫ > 0 there exists aδ > 0 so that ρ(U, Rη(U)) < δ implies |η| < ǫ. Pick N > 0 so that for every x ∈S1,every interval of length δ contains a point of o(x, Rα, N).Now if x1, x2 ∈B satisfydΣ(φ(x1), φ(x2)) < 1/N and if x2 = Rη(x1), then IU = IU ◦Rη when restricted to theset o(x, Rα, N).

Now if ρ(U, Rη(U)) > δ then U△Rη(U) will contain an interval of lengthδ and thus a point of o(x, Rα, N), a contradiction. Thus ρ(U, Rη(U)) < δ and so by thechoice of δ, d(x1, x2) = |η| < ǫ, proving the uniform continuity of φ−1.

Note that φ(B) isdense Gδ in Λ so the extension is almost one to one.Now assume that U is symmetric. The group of numbers r such that Rr(U) = U hasa rational generator, say p/q, with 0 < p/q < 1 and p and q relatively prime.

If U ′ = π(U)where π : S1 →S1/Rp/q is the projection, then Λ(U, α) has Λ(U ′, qα) as a q-fold factor(here we have identified S1/Rp/q with S1). Since U ′ is asymmetric, Λ(U ′, qα) has (S1, Rqα)as a factor, finishing the proof of (2a).9

(2b). Let ψ denote the extension of φ−1 to a continuous semiconjugacy from (Λ, σ) to(S1, Rαq) and assume that m(Fr(U)) = 0.

If λ1 and λ2 are two invariant Borel probabilitymeasures supported on Λ, then since (S1, Rαq) is uniquely ergodic, ψ∗(λ1) = ψ∗(λ2) = m.If X ⊂Λ is a Borel set, then since m(B) = 1, for i = 1, 2, λi(X) = λi(ψ−1(B) ∩X).Now since ψ is injective on B, this is equal to λi(ψ−1(B ∩ψ(X)) = m(B ∩ψ(X)) and soλ1 = λ2.(2c). Now assume Fr(U) is a finite set.

In this case, each x ∈Bc will have exactlytwo preimages under ψ, namely, the limit of φ(xn) as xn →x from the right and the limitof φ(xn) as xn →x from the left. This makes it clear that in this case Λ is conjugateto the minimal set in the circle homeomorphism obtained by “blowing up” into intervalspoints on the orbits of each x ∈Fr(U).(2d).

When U ∈RO0, Λ(U, α) is the support of λ(U, α). Thus to prove (2d) it sufficesto show that Λ(U, α) is an injective function of U.

Assume that for some U1, U2 ∈RO,Λ(U1, α) = Λ(U2, α). Using (2a), φ(B1) and φ2(B2) are dense Gδ in the compact metricspace Λ(U1, α) = Λ(U2, α).

This implies that φ(B1) ∩φ2(B2) ̸= ∅, and so there existx1, x2 ∈S1 with φ1(x1) = φ2(x2).Thus if Rη(x1) = x2, then IU1 = IU2 ◦Rη whenrestricted to the dense set o(x1, Rα). This implies that U1△Rη(U2) contains no intervals.Since the Ui are regular open sets, this means that U1 = Rη(U2) and so U1 and U2 are inthe same equivalence class in RO′, as required.Proof of (3).

Fix p/q ∈Q with p and q relatively prime. Since Rqp/q = Id, it isclear that any s ∈Λ(U, p/q) will satisfy σq(s) = s which implies (3a).

Say a symbol blockb is prime if its length equals its period. For U ∈RO′0, by construction, λ(U, p/q) =P m(Ub,p/q)µb where µb is the probability measure supported on the periodic orbit withrepeating block b and the sum is over all prime blocks b whose period divides q.

With thisformula in hand it is easy to construct a U so that λ(U, p/q) is any desired point in theconvex hull given in the statement of (3b).⊔⊓Proof of Proposition 0.3(a). A theorem of Parthasarathy says that the measures supported on periodic orbitsare dense in M(Σ2, σ) ([P]).

Fix one such measure µ0, and assume it is supported on anorbit of period q. Using the formula given in the proof of Theorem 2 (3b), find a regularopen set U with Fr(U) a finite set and a p/q with λ(U, p/q) = µ0.

Now pick irrationalsαn →p/q. By Theorem 2 (1), λ(U, αn) →µ0, and by Theorem 2 (2c), each λ(U, αn) isthe unique measure supported on a Denjoy minimal set.(b).

It suffices to show that for any symbol block b, there exists an s ∈Σ2 which hasinitial block b and Cl(o(s, σ)) is a Denjoy minimal set. Fix an irrational α and x0 ∈S1.Choose a finite union of intervals U so that Riα(x0) ∈U if and only if bi = 1, for i =0, .

. ., length(b) −1.

Further, the open set U should satisfy o(x0, Rα) ∩Fr(U) = ∅. If Uhas these properties, Theorem 2 (2c) shows that Λ(U, α) is the desired Denjoy minimalset.

⊔⊓Section 3: The Hilbert cube of strictly ergodic sets. We begin with somedefinitions in preparation for the proof of Theorem 0.1.

A copy of the Hilbert cube is given10

by the collection of sequences,H = {γ ∈RN : 0 ≤γi ≤1i + 2 for all i ∈N}.A subspace of H that contains topological balls of all dimensions isH0 = {γ ∈H : γi = 0, for all but finitely many i}.For γ ∈H, define an asymmetric regular open set Uγ byUγ =[i∈N(1i + 2 −γ3i ,1i + 2 + γ3i ).Now define a map Γ : H →RO′0 via Γ(γ) = [Uγ]. It is clear that Γ is continuous andinjective.

Since H is compact, Γ(H) is homeomorphic to H.Proof of Theorem 0.1. Fix an irrational α.

By Theorem 2 (2ab), the set Λ(Γ(H), α)consists of strictly ergodic sets. Since Γ(H) is compact, using Theorem 2 (1) and (2d), wehave that λ(Γ(H), α) is homeomorphic to Γ(H) and therefore to H. This proves the firststatement in the theorem.

To prove the second, note that Theorem 2 (2c) implies thatλ(Γ(H0), α) consists of measures supported on Denjoy minimal sets. Since λ(Γ(H0), α)is homeomorphic to H0, it (and consequently, D(Σ2, σ)) contains topological balls of alldimensions.

⊔⊓Remarks. (3.1) In Theorem 0.1 there is an obvious distinction between S(Σ2, σ), which containsa copy of H, and D(Σ2, σ), which contains a copy of H0.

This is because Λ(Γ(H), α)contains minimal sets that are not Denjoy. In particular, if γ ∈H −H0 and for somei ̸= 0, Riα(0) ∈Fr(Uγ), then Λ(Uγ, α) is not a Denjoy minimal set.

In the semiconjugacyfrom (Λ(Uγ, α), σ) to (S1, Rα), the inverse image of 0 consists of three points.A Denjoy minimal set is obtained from an irrational rotation on the circle by replacing(or ‘blowing up”) each element of a collection of orbits by a pair of orbits. For all γ not ofthe type just described, Λ(Uγ, α) is a Denjoy minimal set.

When γ ∈H0, the number oforbits blown up is the same as the number of distinct orbits containing points of Fr(Uγ).For γ ∈H −H0, if for all i ̸= 0, Riα(0) ̸∈Fr(Uγ), then Λ(Uγ, α) is a Denjoy minimalset with countably many orbits blown up. All the infinite dimensional families we couldconstruct had the property that some minimal set was not Denjoy.

(3.2) Morse and Hedlund’s construction of Sturmian minimal sets corresponds to thespecial case U = (0, α). In this case, Λ(U, α) is a Denjoy minimal set with a single orbitblown up.

(3.3) Theorem 2 (1) states that Γ is a lower semicontinuous function whose range isthe set of closed subsets of a compact metric space. When such functions have a domainthat is a Baire space, they are continuous on a dense, Gδ set (see page 114 of [C]).

It seemsunlikely that RO is a Baire space, but since Γ(H) is homeomorphic to the Hilbert cube,we may apply this result to show that the map (for fixed α)Λ(· , α) : Γ(H) →H(Σ2)11

is continuous at a generic point of Λ(H). This result can also be obtained directly byshowing that the map is, in fact, continuous at all points Γ(γ) for which all points ofFr(Uγ) are on disjoint orbits.

(3.4) As is perhaps obvious from Remark (3.1), when Fr(U) is more complicatedtopologically, so is the structure of Λ(U, α) (for irrational α). However, Theorem 2 (2b)says that for all U ∈RO0, Λ(U, α) is uniquely ergodic.

It is in fact measure isomorphicto (S1, Rα). To get minimal sets with more interesting measure theoretic properties wemust have m(Fr(U)) > 0.

In this case the set BU,α from the main construction is a zeromeasure, dense Gδ set in the circle. This leads one to expect that Λ(U, α) could supportmore than one invariant probability measure.The results of [M-P] show that this is frequently the case.

The relevant constructionfrom that paper begins with a Cantor K in the circle. The complement of K is the disjointunion of open intervals.

One chooses a set of labels for these open sets with each open setlabeled by zero or one. The set of labels is used to construct a minimal set in the two-shiftas in the main construction.

If K has positive measure, then for most sets of labels (in theappropriate sense) the constructed minimal set is not uniquely ergodic and has positivetopological entropy.However, the constructed minimal set can be uniquely ergodic as the following examplesuggested by Benjamin Weiss shows. Let (X, f) be a Denjoy minimal set with intrinsicrotation number α.

Note that (X, f) is both measure isomorphic to and an almost oneto one extension of (S1, Rα). Using results of Jewett and Kreiger we may find a zero-dimensional strictly ergodic system (Z, h) that is mixing and has positive entropy.

Let(Y, g) be the product of the two systems. Because (Z, h) and (X, f) are strictly ergodicand (Z, h) is mixing and (X, f) has pure point spectrum, (Y, g) is strictly ergodic.Now think of Y as an extension of X.

The main theorem and the remark followingTheorem 4 in [F-W] imply that there is a minimal almost 1-1 extension of X , say ( ˜Y , ˜g),which maps onto (X, f) in such a way that the invariant measures of ( ˜Y , ˜g) are in one toone correspondence with the g-invariant measures on Y . Thus ( ˜Y , ˜g) is a strictly ergodic,positive entropy, almost 1-1 extension of rotation by alpha.

Further, as a consequence ofthe method of construction in [F-W], since X, Y , and Z are zero-dimensional, ˜Y is also.Let p : ˜Y →S1 denote the given semiconjugacy and let ˜B ⊂˜Y be the dense Gδ seton which p is injective. Pick two sets, each open and closed, with V0 ⊔V1 = ˜Y .

Note thatU = (p(V0))c is a regular open set. Use the partition {V0, V1} in the usual way to get asymbolic model by defining k : ˜Y →Σ2 so that(k(y))i = IV1(˜gi(y)).It is fairly straightforward to show that ˜B ⊂p−1(BU,α) and thus, p = ψ ◦k where ψ :Λ(U, α) →S1 is the semiconjugacy constructed in the proof of Theorem 2 (2a).

Thisimplies that Λ(U, α) is a factor of ( ˜Y , ˜g), and thus is strictly ergodic. Further, we maychoose V0 and V1 so that Λ(U, α) has positive entropy.

To finish, note that m(Fr(U)) > 0,for if not, Λ(U, α) would be measure isomorphic to the zero entropy system (S1, Rα).It would be interesting to have conditions on a regular open set with positive measurefrontier that distinguish these two cases. More precisely, give necessary and sufficient con-ditions for the unique ergodicity of Λ(U, Rα).

Another interesting question is the structure12

of the set of its invariant measures in the cases when Λ(U, α) is not uniquely ergodic (cf.[Wm]). (3.5) Since each point in S(X, f) represents a disjoint minimal set, the size of S(X, f)should give some indication of the complexity of the dynamics of f. The topological entropyof (X, f), denoted h(X, f), is perhaps the most common way of measuring dynamicalcomplexity.

Corollary 0.2 shows that, at least in some cases, when the topological entropyis positive, S(X, f) is large.If the size of S(X, f) is to give a measure of dynamicalcomplexity, the converse should be true. The next proposition shows that this is not thecase, at least when the “size” of S(X, f) is measured by the maximal dimension of anembedded ball and X is a manifold of dimension greater than two.Proposition 3.1.

(a) There exists a compact shift invariant set ˆΛ ⊂Σ2 such that S(ˆΛ, σ) is homeomor-phic to the Hilbert cube and h(ˆΛ, σ) = 0. (b) On any smooth manifold M with dimension greater than two there exists a C∞diffeomorphism f such that h(f) = 0 and S(M, f) contains a subspace homeo-morphic to the Hilbert cube.Proof of (a).Fix an irrational α and let T = S1 × H.Define F : T →T asF = Rα × Id.

We will do a construction analogous to the main construction, but nowusing the space T and the map F. To get an open set in T we use the open sets Uγconstructed above to defineˆU =[γ∈HUγ × {γ}.Next letˆB = {β ∈T : o(β, F) ∩Fr( ˆU) = ∅}and define Φ : ˆB →Σ2 so that(Φ(β))i = I ˆU(F i(β)).Finally, let ˆΛ = Cl(Φ( ˆB)).Note that for fixed γ, Φ restricted to (S1 × {γ}) ∩ˆB is just φUγ,α from the mainconstruction and thatˆΛ = Cl([γ∈HΛ(Uγ, α)).Theorem 2 (2a) and (2d) imply that Φ is injective. Using an argument similar to onein the proof of Theorem 2 (2a), one gets that Φ−1 is uniformly continuous, and thereforehas a continuous extension to a Ψ : ˆΛ →T that satisfies Ψ ◦σ = F ◦Ψ.The variational principle (see page 190 in [W]) implies that h(ˆΛ, σ) = 0 if all ergodicmeasures for (ˆΛ, σ) have metric entropy zero.

If η is an ergodic, invariant Borel probabilitymeasure for ˆΛ, then Ψ∗(η) is such a measure for (T, F) and so Ψ∗(η) is Haar measure onS1 × {γ0} for some γ0. This implies that η is supported on Ψ−1(S1 × {γ0}).

Once again,using an argument virtually identical to one in the proof of Theorem 2 (2b), one obtainsη = λ(Uγ0, α). This measure with the shift is measure isomorphic to rotation on the circle13

by α and therefore has zero metric entropy, as required. Note that the argument just givenalso shows that S(ˆΛ, σ) is in fact homeomorphic to the Hilbert cube, H.Proof of (b).

We first construct the map on the space P = D2 × [−1, 1], whereD2 is a closed two-dimensional disk. Let h : D2 →D2 be a Smale horseshoe, i.e.

h isa C∞-diffeomorphism whose nonwandering set consists of the union of a finite number offixed points and a set Ωon which the dynamics are conjugate to the full two-shift. Thecompact invariant set ˆΛ constructed in the proof of (a) is embedded in Ωby the conjugacy.Call this embedded set ¯Λ.Next, let ht for t ∈[−1, 1] be an isotopy with h−1 = Id, h0 = h, and h1 = Id.Further, ht restricted to the boundary of D2 should be the identity for all t. Now pick aC∞-function w : P →R with w ≥0 and w−1(0) = ∂P ⊔(¯Λ × {0}).

Let g : P →P bethe time one map of the flow generated by the vector field w(u) ∂∂z , where u = (x, y, z) isa point in P. Now let f = g ◦(ht × Id). By construction, the nonwandering set of f is∂P ⊔(¯Λ × {0}) and thus h(f) = 0.

Since each point on ∂P is a fixed point for f, S(P, f)is homeomorphic to S(¯Λ, σ) ⊔∂P, which in turn, is homeomorphic to H ⊔∂P.To obtain the result on a general manifold of dimension three or higher, embed a copyof (P, f) in it and extend f by the identity on the rest of the manifold. ⊔⊓Remarks(3.6) This proposition leaves open the possibility of a converse to Corollary 0.2 indimension 2.

In this dimension there are a number of results that show that the existenceof certain types of zero entropy invariant sets can imply that a homeomorphism has positivetopological entropy. For example, if an orientation-reversing homeomorphism of a compactsurface of genus g has periodic orbits with g + 2 distinct odd periods, then it has positiveentropy ([B-F], [H]).

For orientation-preserving homeomorphisms there are restrictions onthe periods that occur in zero entropy maps given in [S]. Even a single period orbit canimply positive entropy if the isotopy class on its complement is nontrivial ([Bd]).

Theseresults give credence to the conjecture that for a manifold M of dimension 2, if f : M →Mis a homeomorphism and S(M, f) contains a topological ball of dimension 3, then h(f) > 0. (3.7) It was noted in the introduction that the existence of a Hilbert cube of strictlyergodic sets can often be viewed as a manifestation of a standard topological fact, namely,the Hilbert cube is the continuous surjective image of the Cantor set.

For concreteness, letf : M →M be a homeomorphism with an invariant set ¯Λ with (¯Λ, f) conjugate to (ˆΛ, σ),where ˆΛ is the set constructed in the proof of Proposition 3.1 (b). Using the conjugacy, theproof of Proposition 3.1 (b), and Theorem 2 (2b) one gets that for each x ∈¯Λ, Cl(o(x, f))supports a single invariant probability measure which is c∗(λ(Uγ(x), α)) for the appropriateγ(x).

Further, the map x 7→c∗(λ(Uγ(x), α)) is continuous. (More formally, this map isx 7→c∗(λ(Γ(π2(Ψ(x))), α))where π2 : S1 ×H →H is the projection).

The domain of this map is the invariant Cantorset ¯Λ and its image is λ(Γ(H), α), which is homeomorphic to the Hilbert cube, H.(3.8) The construction in the proof of 3.1 (b) can be used to embed any compactshift invariant subset of Σ2 as the only “interesting” dynamics in a three-dimensionaldiffeomorphism. It is reminiscent of Schweitzer’s construction of C1-counterexample tothe Seifert conjecture ([Sc]).14

Section 4: Intrinsic and extrinsic rotation numbers. In the Section 1 it wasnoted that abstract Denjoy minimal sets have well-defined intrinsic rotation numbers.

Thenext proposition specializes some previous results to the case of fixed intrinsic rotationnumber.Proposition 4.1. Fix an irrational α and let Dα(Σ2, σ) denote the set of Denjoyminimal sets in the shift with intrinsic rotation number α.

(a) When given the weak topology, the space Dα(Σ2, σ) contains topological balls ofdimension n for all natural numbers n.(b) The set of points that are members of Denjoy minimal sets with intrinsic rotationnumber α is dense in Σ2. (c) If (D, σ) is a Denjoy minimal set with intrinsic rotation number α, then D =Λ(U, α) for some regular open set U with m(Fr(U)) = 0.

Consequently, Dα(Σ2, σ)⊂λ(RO0, α).Proof of Proposition 4.1. When U is asymmetric and Λ(U, α) is a Denjoy minimalset, it has intrinsic rotation number α.

This follows from Theorem 2 (2b) (and its proof).Thus to prove (a) we need only note that the proof of Theorem 0.1 began with a statement,“ Fix an irrational α”. The proof of Proposition 0.3 (b) contains a similar statement, sothat proof proves (b).To prove (c), note that by definition, there exists a conjugacy c : D →Y where Y isthe minimal set in a Denjoy example g : S1 →S1 with rotation number α.

It is a standardfact that there exists a semiconjugacy h of (S1, g) to (S1, Rα) with the properties that h isinjective on a set that is dense in Y and the lift of h is weakly order preserving, i.e. x < yimplies ˜h(x) ≤˜h(y).Now let p = h◦c and U = (p(C0))c. Since C0 is compact in Σ2, U is open.

Further, theproperties given above imply that U ∗= (p(C1))c and p(C0) ∩p(C1) = Fr(U) = Fr(U ∗).Thus using a fact from Section 1, U is a regular open set, and by construction, Λ(U, α) = D.Since p(C0) ∩p(C1) is at most countable, m(Fr(U)) = 0.⊔⊓These results, of course, also hold for homeomorphisms with a full two-shift embeddedin their dynamics.In this case, however, one is perhaps more interested in extrinsicproperties of invariant sets, i.e. properties associated with how the sets are embeddedin the manifold.

Perhaps the simplest such extrinsic property is the extrinsic rotationnumber, and the simplest case in which this can be defined is for a homeomorphism of theannulus.If f : A →A is a homeomorphism of the annulus and z ∈A, define the rotationnumber of z under f asρ(z) = limn→∞π1( ˜f n(˜z)) −π1(˜z)n,if the limit exists. Here ˜f : R×[−1, 1] →R×[−1, 1] and ˜z are lifts of f and z, respectively,and π1 : R × [−1, 1] →R is the projection.

Note that the rotation number is only definedmodulo 1 as it depends on the choice of lift.If D ⊂A is a Denjoy minimal set under f, then it is uniquely ergodic. Thus for allz ∈D, ρ(z) =Rr(z) dµ, where µ is the the unique invariant probability measure of (D, f)and r : S1 →R is the map that lifts to π1 ◦˜f −π1.

This number will be called the extrinsicrotation number of (D, f).15

The Denjoy minimal sets constructed by Mather in [M] have monotonicity propertiesthat imply that their extrinsic and intrinsic rotation numbers are rationally related. ForDenjoy minimal sets in a general homeomorphism of the annulus this will not be the case.As a specific example, we will consider homeomorphisms f : A →A that have a rotaryhorseshoe (cf.

[H-H2]) A picture of the lift of such a map is shown in Figure 1. The dottedvertical lines are the boundaries of fundamental domains.010101fFigure 1: The lift of a rotary horseshoe.A map contains a rotary horseshoe if it has a compact invariant set Ωthat is conjugateto the full two-shift.

The conjugacy c : Ω→Σ2 is required to have the property that forz ∈Ωthe first element in c(z) is 1 if and only if ˜f moves ˜z (approximately) one fundamentaldomain to the right. More precisely, for z ∈Ωit is required thatρ(z) = limN→∞NXi=0IC1(σi(c(z)))(N + 1).Thus ρ(z) is the asymptotic average number of ones in the sequence c(z).We are now almost in a position to state a result about the existence of Denjoy minimalsets with given intrinsic and extrinsic rotation number.

For an annulus homeomorphismf, let Dα,β(A, f) denote the set of all Denjoy minimal sets for f with intrinsic rotationnumber α and extrinsic rotation number β.Proposition 4.2. If a homeomorphism f : A →A has a rotary horseshoe, then forall irrational α, and all β ∈S1, Dα,β(A, f) contains topological balls of dimension n forall natural numbers n.Proof of Proposition 4.2.

If for a given U ∈RO0 and irrational α, Λ(U, α) is aDenjoy minimal set, then the comments above Lemma 1 and unique ergodicity imply thatfor all s ∈Λ(U, α),limN→∞NXi=0IC1(σi(s))(N + 1)= λ(U, α)[C1] = m(U).16

This implies that the corresponding Denjoy minimal set in the annulus has extrinsic ro-tation number equal to m(U). To finish the proof, one need only imitate the proof ofTheorem 0.1 using a family Uγ that satisfies m(Uγ) = β, for all γ.

⊔⊓Note that the case of rational β is included in this result. This means that largedimensional balls of Denjoy minimal sets with a given rational extrinsic rotation numberare present in the dynamics.REFERENCES[A] Auslander, J., Minimal Flows and their Extensions, North Holland Mathematics Stud-ies, vol.

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