Zero Measure Spectrum for Multi-Frequency Schrödinger Operators
aa r X i v : . [ m a t h . SP ] S e p ZERO MEASURE SPECTRUM FOR MULTI-FREQUENCYSCHR ¨ODINGER OPERATORS
JON CHAIKA, DAVID DAMANIK, JAKE FILLMAN, AND PHILIPP GOHLKE
Abstract.
Building on works of Berth´e–Steiner–Thuswaldner and Fogg–Nous we show that on the two-dimensional torus, Lebesgue almost everytranslation admits a natural coding such that the associated subshift sat-isfies the Boshernitzan criterion. As a consequence we show that for thesetorus translations, every quasi-periodic potential can be approximated uni-formly by one for which the associated Schr¨odinger operator has Cantorspectrum of zero Lebesgue measure. We also describe a framework thatcan allow this to be extended to higher-dimensional tori. Introduction
This work addresses the persistent occurrence of Cantor spectrum of zeroLebesgue measure in the class of discrete one-dimensional Schr¨odinger opera-tors with generalized quasi-periodic potentials, where the underlying torus hasdimension strictly greater than one.To motivate this problem, let us describe the setting and recall some of theknown results. Fix a dimension d ∈ N and consider α ∈ T d := R d / Z d that issuch that the translation R α : T d → T d , ω ω + α is minimal. If g : T d → R is bounded and measurable, we can consider, for each ω ∈ T d , the discreteSchr¨odinger operator[ H α,g,ω ψ ]( n ) = ψ ( n + 1) + ψ ( n −
1) + g ( ω + nα ) ψ ( n )in ℓ ( Z ). We call such an operator a generalized quasi-periodic Schr¨odingeroperator . Within this class of sampling functions, one distinguishes severalstandard regularity classes and observes that the spectral properties of theoperators in question depend quite significantly on the chosen regularity class.Standard examples are given by continuous g (this corresponds precisely to J.C. was supported in part by the Simons foundation, the Warnock chair, and NSF GrantDMS–1452762.D.D. was supported in part by NSF grant DMS–1700131 and by an Alexander von Hum-boldt Foundation research award.J.F. was supported in part by Simons Collaboration Grant the class of quasi-periodic Schr¨odinger operators ), H¨older continuous g , g thatare differentiable a certain finite number of times, smooth (i.e., infinitely dif-ferentiable) g , and analytic g .One is interested in the spectrum and the spectral type. By standard argu-ments involving the ergodicity of Lebesgue measure with respect to R α , thereis a compact set Σ α,g such that for Lebesgue almost every ω ∈ T d , the spec-trum of H α,g,ω is equal to Σ α,g . Similarly, the spectral type of H α,g,ω is alsoLebesgue-almost surely independent of ω . As we will focus on the spectrum inthis paper, we will not go into further details regarding the spectral type andrefer the reader to the surveys [12, 23] for background and more information.The almost sure spectrum Σ α,g can have various topological and measure-theoretic properties. It can be a Cantor (i.e., perfect and nowhere dense) set,but it can also be a finite union of non-degenerate compact intervals. TheCantor spectra that occur can have both positive and zero Lebesgue measure.Among those that have zero Lebesgue measure, examples are known withsmall, and even zero, Hausdorff dimension.Roughly speaking, when d = 1, it is well known how to produce exampleswith zero Lebesgue measure [13, 14] and even zero Hausdorff dimension [22].On the other hand, when d >
1, examples are known where the spectrum is afinite union of intervals, and it is (essentially) open how to produce spectra ofzero Lebesgue measure. The present paper develops a way of producing manysuch examples. Indeed they are “ample” in a way we will make precise.Since we used zero Lebesgue measure and non-Cantor structure to distin-guish between the two cases d = 1 and d > C ( T d ) forany fixed minimal translation R α (without supplying any information aboutthe Lebesgue measure of the set) has a proof that works simultaneously forall values of d ∈ N ; see [4, 5]. On the other hand, in the analytic category,Cantor spectrum is typical when d = 1 (the literature is extensive; see, e.g.,[16, 18, 24], and the surveys [12, 23] for a more complete list), while it is nottypical when d > d = 1), but so far they arepoorly understood in the multi-frequency case ( d > There is a way to recast some known results for primitive substitution subshifts interms of codings of torus translations; see, for example, [25] for the case of the Tribonaccisubstitution and [1] for more examples.
ERO MEASURE SPECTRUM FOR MULTI-FREQUENCY OPERATORS 3 of examples that is in some ways as rich and “ample” as the existing work inthe one-frequency case.
Definition 1.1.
A function g : T d → R is called elementary if it is measurableand takes finitely many values. The set of elementary functions g : T d → R is denoted by E ( T d ). A subset of E ( T d ) is called ample if its k · k ∞ -closure in L ∞ ( T d ) contains C ( T d ). Theorem 1.2.
Let d = 2 . Then, for Lebesgue almost every α ∈ T d , the set Z α = { g ∈ E ( T d ) : Σ α,g is a Cantor set of zero Lebesgue measure } is ample. Remark 1.3. (a) In the case d = 1, this is a result of Damanik-Lenz [13, 14].Specifically, it follows by combining [13, Theorem 2] and [14, Theorem 10].Actually, in this case, the full measure set of α ∈ T is explicit: it is the set ofall irrational numbers. By contrast, the full measure set in Theorem 1.2 is notexplicit.(b) The fact that the result can be extended to a value of d that is greaterthan one is not obvious, and indeed surprising, since the straightforward ex-tension of [14, Theorem 10] is known to fail, compare Remark 3.7 below.(c) The proof of Theorem 1.2 also employs [13, Theorem 2], but replacesthe use of [14, Theorem 10] by a more sophisticated process to verify theassumption of [13, Theorem 2].(d) To the best of our knowledge, there is no known example of a quasi-periodic multi-frequency potential (i.e., d > g ∈ C ( T d )) so that the asso-ciated Schr¨odinger operator has zero-measure spectrum. It is unclear whethersuch an example exists. The fact that arbitrarily small k · k ∞ perturbations ofan arbitrary g ∈ C ( T d ) can produce this effect is therefore interesting.(e) We described the occurrence of zero-measure spectrum obtained via thisroute as “persistent” above, so let us explain what we mean by that. The g ∈ E ( T ) we obtain for which Σ α,g is a Cantor set of zero Lebesgue measureare actually such that Σ α,λg is a Cantor set of zero Lebesgue measure for every λ ∈ R with λ = 0. Thus the phenomenon is persistent with respect to varyingthe coupling constant. This should be contrasted with the fact that any known g ∈ C ( T ) for which Σ α,g has been shown to have zero Lebesgue measure forsuitable (irrational) α ∈ T is such that Σ α,λg has positive Lebesgue measurefor every λ ∈ R with | λ | 6 = 1. In other words, the zero-measure property ishighly unstable with respect to a variation of the coupling constant in thequasi-periodic setting.(f) We regard it as an interesting open problem to explore whether Theo-rem 1.2 can be extended to some larger values of d . Several components of J. CHAIKA, D. DAMANIK, J. FILLMAN, AND P. GOHLKE our proof of Theorem 1.2 indeed do extend to values of d greater than 2. Inthe final section of this paper we comment on why our result is limited to thecase d = 2 and point out the obstacles one needs to overcome if one wants toprove a result for some d > d ≥ Preliminaries
Multi-Dimensional Continued Fraction Algorithms.
Motivation and Notation.
Continued fractions are a tool to understandthe Diophantine properties of numbers and the dynamical properties of rota-tions. The theory has been best developed in dimension one where the Eu-clidean algorithm and its acceleration, the Gauss map, are incredibly useful.There are many generalizations of these algorithms to higher dimensions. Forour purposes we will restrict our attention to the Cassaigne-Selmer algorithmand the Brun algorithm (the latter in the special case of four dimensions).2.1.2.
The Cassaigne-Selmer Algorithm.
Denote R + = [0 , ∞ ) and let∆ = ∆ = { ( x , x , x ) ∈ R : x + x + x = 1 } . The
Cassaigne-Selmer algorithm is given by(2.1) T C : ∆ → ∆ by T ( x , x , x ) = ( ( x − x x + x , x x + x , x x + x ) if x ≥ x ( x x + x , x x + x , x − x x + x ) if x > x . This algorithm was studied in [10] for its connection to word combinatorics.There is an ergodic T C -invariant probability measure ν C on ∆ which is equiv-alent to Lebesgue measure. Indeed, the Cassaigne-Selmer algorithm is conju-gate to the Selmer algorithm [10]. This algorithm is ergodic by [26, Section 7],whose argument presenting the proof of ergodicity of the fully sorted Selmeralgorithm generalizes to show that the semi-sorted Selmer algorithm is ergodic. ERO MEASURE SPECTRUM FOR MULTI-FREQUENCY OPERATORS 5
The Brun Algorithm for d = 4 . Let∆ = ∆ = { ( x , x , x , x ) ∈ R : x + x + x + x = 1 } and, for i, j ∈ { , , , } , let∆( i, j ) = { ( x , x , x , x ) : x i ≥ x j ≥ x k for all k / ∈ { i, j }} . The
Brun algorithm T B : ∆ → ∆ is defined for ( x , . . . , x ) ∈ ∆( i, j ) as T B ( x , . . . , x ) k = x k − x j if k = i, x i − x j − x j if k = i. This map is well-defined almost everywhere on ∆. The ergodicity of this algo-rithm follows as in [26]. Hence, there exists an ergodic T B -invariant probabilitymeasure ν B on ∆, which is equivalent to Lebesgue measure.2.2. S-Adic Subshifts.
Given a finite set A , give the full shift A Z the producttopology inherited from placing the discrete topology on each factor, and definethe shift map S : A Z → A Z by [ Sx ]( n ) = x ( n + 1). A subshift over A is aclosed (hence compact) S -invariant subset X ⊆ A Z .The free monoid will be denoted A ∗ = S ∞ n =0 A n ; the unique element of A is denoted ε and called the empty word; the length of u ∈ A n is | u | = n . Write(2.2) u ( v ) := { j : v j +1 v j +2 · · · v j + | u | = u } for the number of times u occurs in v , u ⊳ v if u ( v ) >
0, and L ( u ) for the setof all subwords of u ∈ A ∗ , A N or A Z . For a subshift X , the language of X is L ( X ) := { u : u ∈ L ( x ) for some x ∈ X } . When (
X, S ) is minimal, L ( X ) = L ( x ) for every x ∈ X . Definition 2.1.
Let (
X, S ) be a minimal subshift. We say that (
X, S ) satisfiesthe
Boshernitzan criterion if there exist an S -invariant probability measure µ , a constant C >
0, and a sequence n , n , . . . → ∞ so that for all w = w · · · w n i ∈ L ( X ), µ ( { x ∈ X : x · · · x n i = w } > Cn i . A substitution is an endomorphism τ : A ∗ → A ∗ , which is uniquely definedby its values on individual letters of A . We shall also assume that all substi-tutions are non-erasing in the sense that τ ( a ) = ε for every a ∈ A , and denotethe set of non-erasing substitutions on A by Sub( A ). For each τ ∈ Sub( A ),one associates the substitution matrix M = M τ ∈ End( Z A ), with entries givenby M τ [ a, b ] = a ( τ ( b )) . J. CHAIKA, D. DAMANIK, J. FILLMAN, AND P. GOHLKE An S-adic system over A is defined by a choice of a directive sequence τ =( τ n ) ∞ n =0 of substitutions on A . We will encounter products quite frequently,so, for 0 ≤ m < n , we write τ [ m,n ] = τ m · · · τ n , with obvious conventions for open and half-open intervals. For a ∈ A , write w n ( a ) = τ [0 ,n ] ( a ). Similarly, for the substitution matrices, we write M I = M τ I for an interval I . Clearly, for I = [ m, n ], one has M [ m,n ] = M τ m M τ m +1 · · · M τ n . The language associated to τ is L ( τ ) := { w ∈ A ∗ : w ⊳ w n ( a ) for some a ∈ A and n ∈ N } . We also call this the set of allowed words. It is easy to check that X = X ( τ ) := (cid:8) x ∈ A Z : L ( x ) ⊆ L ( τ ) (cid:9) , is a non-empty subshift, provided thatlim n →∞ max a ∈A | w n ( a ) | = ∞ . In this case, we call X ( τ ) the S-adic subshift generated by τ .2.3. S-Adic Subshifts Related to Multi-Dimensional Continued Frac-tions.
Both the Cassaigne-Selmer algorithm and the Brun algorithm are ofthe form T : ∆ → ∆ , x A ( x ) − x k A ( x ) − x k for some locally constant matrix valued function A : ∆ → GL( d, Z ). Following[8], we select for each x ∈ ∆ a substitution ϕ ( x ) on the alphabet A = { , . . . , d } such that A ( x ) coincides with the substitution matrix M ϕ ( x ) . In the case ofthe Cassaigne-Selmer algorithm this is achieved by ϕ ( x ) = ( γ if x ≥ x ,γ if x > x , with the Cassaigne-Selmer substitutions γ : γ : . For the Brun algorithm we consider the class of substitutions β ij : j ij, k k for k ∈ A \ { j } . for i, j ∈ A = { , , , } and we set ϕ ( x ) = β ij for x ∈ ∆( i, j ). ERO MEASURE SPECTRUM FOR MULTI-FREQUENCY OPERATORS 7
Given a substitution selection ϕ : ∆ → Sub( A ), the orbit of a point x ∈ ∆under the action of T defines an S-adic system, called a substitutive realization of (∆ , T, A ), given by the directive sequence φ ( x ) = ( ϕ ( T n x )) ∞ n =0 . The corresponding subshift is given by ( X ( φ ( x )) , S ). On the other hand, werelate to each point x in the d -dimensional simplex ∆ a point on the torus T d − by the map π : ∆ → T d − , which denotes the projection to the first d − π is not a surjective map but for T d − = { t ∈ T d − : t + . . . + t d − ≤ } , the map π : ∆ → T d − , x π ( x ) is a bijection, identifying T d − ∼ = [0 , d − inthe obvious fashion. Slightly abusing notation, we use the same symbol, π , todenote both maps.2.4. Natural Codings of Torus Translations.
For the d -dimensional torus T d and α ∈ T d , let R α : T d → T d , R α ( ω ) = ω + α denote the torus translationassociated to α .We present in the following a weaker version of the term natural coding as defined in [8]. This turns some of the results we cite from [8] into merecorollaries which are, however, sufficient for our purposes. A collection F = {F , . . . , F h } is called a natural measurable partition of T d if S hi =1 F i = T d , F j ∩ F k has zero measure for each j = k , and each F i is measurable withdense interior and zero measure boundary. Given the map R α , the languageassociated with F , denoted L ( F ), is the set of finite words w = w · · · w n ∈{ , . . . , h } ∗ such that T nk =0 R − kα ˚ F w k = ∅ , where ˚ A denotes the interior of A . Definition 2.2.
A subshift (
X, S ) is called a natural coding of ( T d , R α ) ifits language coincides with the language of a natural measurable partition {F , . . . , F h } and \ n ∈ N n \ k =0 R − kα ˚ F x k consists of a single point for every x = ( x n ) n ∈ Z ∈ X .The following result concerning the Cassaigne-Selmer algorithm is essentialfor our analysis. Proposition 2.3. [8, Theorem 6.2]
Let φ be the substitutive realization ofthe Cassaigne-Selmer algorithm. For ν C -almost every x ∈ ∆ , the subshift ( X ( φ ( x )) , S ) is a natural coding of ( T , R π ( x ) ) . Note that [17, Theorems A and B] are closely related results, that wouldhave also been sufficient for our purposes.
J. CHAIKA, D. DAMANIK, J. FILLMAN, AND P. GOHLKE
Remark 2.4. If F = {F , . . . , F h } is a natural measurable partition of T and M ∈ GL(2 , Z ), then the language generated by R α on F coincides withthe language generated by R Mα on the natural measurable partition M F = { M F , . . . , M F h } . In particular, if ( X, S ) is a natural coding of ( T , R α ), thenit is also a natural coding of ( T , R Mα ). In two dimensions we could simplytake M α := − α , to obtain natural codings for (almost) all α ∈ T from codingsfor α ∈ T . For the more general d -dimensional cases, we still obtain T d from T d ∆ via general linear transformations, compare [8, Rem. 3.5].One would naturally like to obtain analogs of Proposition 2.3 for higher-dimensional torus translations. For such translations, the Brun algorithm isa natural candidate to use for the associated continued fraction algorithm.However, in that case, there is a technical ingredient (namely negativity ofthe second Lyapunov exponent) which is currently unclear. We discuss this inmore detail in Section 6.2.5. Zero-Measure Spectrum via the Boshernitzan Criterion.
Givena finite alphabet A and a subshift X ⊆ A Z , one can define Schr¨odinger opera-tors in ℓ ( Z ) by generating potentials which are obtained through real-valuedsampling along the S -orbits of X . That is, if f : X → R is given, we associatewith each x ∈ X the potential V x : Z → R given by V x ( n ) = f ( S n x ), n ∈ Z .The Schr¨odinger operator H x in ℓ ( Z ) is then given by[ H x ψ ]( n ) = ψ ( n + 1) + ψ ( n −
1) + V x ( n ) ψ ( n ) . One typically restricts attention to locally constant functions f , that is, func-tions that depend on only finitely many entries of the input sequence x . Suchfunctions are of course continuous, but in addition they preserve the finite-valuedness, which is crucial to many arguments in the study of these operators.If X is minimal and f is locally constant, then a simple strong approximationargument shows that there is a compact set Σ X,f ⊂ R such that σ ( H x ) = Σ X,f for every x ∈ X . Obviously, a minimal subshift X is finite if and only if every V x is periodic, and in this case Σ X,f is well known to be a union of finitely manynon-degenerate compact intervals. Similarly, if f is constant, the same con-clusions hold. Ruling out these degenerate cases, it is an interesting questionwhether Σ X,f must have zero Lebesgue measure. In fact, Simon conjecturedthat this must be the case in complete generality, but this conjecture has beendisproved in [6].On the other hand, the Boshernitzan criterion turns out to be a sufficientcondition [13, Theorem 2]:
ERO MEASURE SPECTRUM FOR MULTI-FREQUENCY OPERATORS 9
Theorem 2.5.
If the minimal subshift X satisfies the Boshernitzan criterionand f is locally constant, then either all V x are periodic or the set Σ X,f is aCantor set of zero Lebesgue measure. S-Adic Subshifts Satisfying the Boshernitzan Criterion
Let τ = ( τ k ) ∞ k =0 be a directive sequence generating an S-adic system,( X ( τ ) , S ). Refer to Section 2.2 for definitions and notation. Our key auxil-iary result is a sufficient criterion on τ for ( X ( τ ) , S ) to satisfy Boshernitzan’scriterion for unique ergodicity. Definition 3.1.
For a, b ∈ A , we say that a precedes b at level n if there are m ∈ N and c ∈ A such that ab ⊳ τ [ n +1 ,n + m ] ( c ). For an interval I = [ n + 1 , n + ℓ ],we say τ I is a word builder at level n if, whenever a precedes b at level n , thereis c ∈ A such that ab ⊳ τ I ( c ). Theorem 3.2.
Suppose there exists a constant
N > so that, for infintelymany n , there exist n < n < n < n so that (a) M [ n +1 ,n ] and M [ n +1 ,n ] are positive matrices, (b) τ [ n +1 ,n ] is a word builder at level n , (c) max {k M [ n +1 ,n ] k , k M [ n +1 ,n ] k , k M [ n +1 ,n ] k} ≤ N .Then ( X ( τ ) , S ) satisfies Boshernitzan’s criterion. Lemma 3.3. If τ n +1 ( a ) = b . . . b r , then w n +1 ( a ) = w n ( b ) . . . w n ( b r ) .Proof. This follows immediately from the definition of w n . (cid:3) Corollary 3.4.
Let n, k ∈ N . If M [ n,n + k ] is a positive matrix, then, for all a, a ′ ∈ A , one has | w n + k ( a ) || w n + k ( a ′ ) | ≤ max i,j,j ′ (cid:26) M [ n,n + k ] [ i, j ] M [ n,n + k ] [ i, j ′ ] (cid:27) . Proof.
For each b ∈ A , we apply Lemma 3.3 k times to write w n + k ( b ) as aconcatenation of w n ( a ) for a ∈ A . For each i, j, j ′ ∈ A , the ratio of occurrencesof w n ( i ) in such a decomposition of w n + k ( j ) and w n + k ( j ′ ) is at most the righthand side. (cid:3) Lemma 3.5. If τ [ n +1 ,n + ℓ ] is a word builder at level n , then every allowed wordof length at most min c ∈A | w n ( c ) | is a subword of w n + ℓ ( c ) for some c ∈ A .Proof. Every word is a truncation of concatenations of w n ( c ) as c varies in A . So every word of length at most min c ∈A | w n ( c ) | is formed by concatenatinga (possibly empty) suffix of w n ( a ) with a (possibly empty) prefix of w n ( a ′ ) where a precedes a ′ at level n . All such combinations appear in w n + ℓ ( c ) forsome c ∈ A . (cid:3) Lemma 3.6. If τ [ n +1 ,n + ℓ ] is a word builder and M [ n + ℓ +1 ,n + ℓ + k ] is positive,then the measure of the cylinder set associated with any word of length min a ∈A | w n ( a ) | is at least (cid:18) max c ∈A | w n + ℓ + k ( c ) | (cid:19) − . Proof.
Every allowed word of length at most min a ∈A | w n ( a ) | appears at least oncein every w n + ℓ + k ( c ). Indeed, every c appears in τ n + ℓ +1 · · · τ n + ℓ + k ( a ) by the posi-tivity of M n + ℓ +1 · · · M n + ℓ + k . So every w n + ℓ ( c ) appears in every w n + ℓ + k ( a ). ByLemma 3.5 this implies that every allowed word of length at most min a ∈A | w n ( a ) | appears at least once in every w n + ℓ + k ( c ).So the proportion of every allowed word in such blocks is at least(max c ∈A | w n + ℓ + k ( c ) | ) − . As our language is a concatenation of w n + ℓ + k ( c ) as c varies in A we have the claim. (cid:3) Proof of Theorem 3.2.
This follows from Lemma 3.6 and Corollary 3.4. In-deed, max c ∈A | w n ( c ) | ≤ N (cid:18) max c ∈A | w n ( c ) | (cid:19) ≤ N (cid:18) min c ∈A | w n ( c ) | (cid:19) . So we have that the measure of any cylinder of length min c ∈A | w n ( c ) | is at least( N min c ∈A | w n ( c ) | ) − . Consequently, there exist infinitely many r so that wesatisfy the Boshernitzan criterion with C = ( N r ) − . (cid:3) Remark 3.7.
It is easy to see that any subshift satisfying the Boshernitzancriterion must have a complexity function that is linearly bounded on a subse-quence. This in turn shows that for codings of higher-dimensional torus trans-lations, care must be taken if there is to be any hope to generate subshiftssatisfying the Boshernitzan criterion. Indeed, it is known that any coding of aminimal translation of T d , d ≥
2, relative to a partition of T d into sufficientlynice sets has a super-linear lower bound; compare, for example, [11, 29]. ERO MEASURE SPECTRUM FOR MULTI-FREQUENCY OPERATORS 11
2D Toral Translations
The substitution matrices associated to the Cassaigne–Selmer substitutions γ and γ are given by C = and C = , respectively. Recall that ν C denotes the T C -ergodic measure on ∆ which isequivalent to Lebesgue measure. For the remainder of this section, let T = T C and ν = ν C . The pushforward of Lebesgue measure on ∆ under π isequivalent to Lebesuge measure (and therefore to ν ) on T . Hence, for almostall α ∈ T , the subshift ( X ( φ ( π − ( α ))) , S ) is a natural coding of ( T , R α ) dueto Proposition 2.3. Proposition 4.1.
For Lebesgue a.e. α ∈ T , the subshift ( X ( φ ( π − ( α ))) , S ) satisfies Boshernitzan’s criterion. In particular, for almost every α ∈ T , thetoral translation ( T , R α ) admits a natural coding that satisfies Boshernitzan’scriterion.Proof. It suffices to show that for ν -almost every x ∈ ∆, the subshift( X ( φ ( x )) , S ) satisfies Boshernitzan’s criterion. Note that τ = γ ◦ γ is aprimitive substitution, indeed M τ is positive.Further we claim that the substitution τ ′ = γ γ γ γ γ is a word builder,irrespective of its position within a directive sequence ( τ n ) ∞ n =0 ∈ { γ , γ } N . Toverify this, we first observe that the set γ ( A ) does not contain any of thewords in { , , , } as a subword. Hence, whenever τ ′ = τ [ n +1 ,n +8] and a precedes b at level n , it follows that ab ∈ L := { , , , , } . A directcalculation yields that τ ′ (1) = 1213113 and so for all ab ∈ L we find that ab ⊳ τ ′ (1). In particular, τ ′ is a word builder. The substitution τ ∗ = τ τ ′ τ isa composition of ℓ = 14 substitutions drawn from { γ , γ } . Let B m = { ( τ n ) n ∈ N ∈ { γ , γ } N : τ m ◦ . . . ◦ τ m + ℓ − = τ ∗ } and B = lim m →∞ B m . By Theorem 3.2, for every τ ∈ B , the correspondingsubshift ( X ( τ ) , S ) satisfies Boshernitzan’s criterion. Hence, it is enough toshow that µ = ν ◦ φ − assigns full measure to B . We consider the set D = φ − ( B ) = { x ∈ ∆ : A ( x ) · · · A ( T ℓ − x ) = M τ ∗ } Since the map φ conjugates T and S , we have that φ − ( B m ) = φ − ( S − m B ) = T − m φ − ( B ) = T − m D for all m ∈ N . By Birkhoff’s ergodic theorem, we have for almost every x ∈ ∆that lim n →∞ n n − X m =0 φ − ( B m ) ( x ) = lim n →∞ n n − X m =0 D ( T m x ) = ν ( D ) . If ν ( D ) >
0, we therefore conclude that almost-every x is contained in infinitelymany φ − ( B m ) and hence in φ − ( B ), implying ν ( φ − ( B )) = 1. It remains toshow that ν ( D ) > { x ∈ ∆ : x ≥ x } and ∆(2) = { x ∈ ∆ : x > x } , that is, A ( x ) = C i ⇐⇒ x ∈ ∆( i ) . In the following, we identify sets that coincide up to a set of Lebesgue measurezero—this applies in particular to the boundaries of the sets ∆, ∆(1) and∆(2). Since T (∆( i )) = ∆ and T acts on ∆( i ) as the radial projection of C − i (∆( i )) to ∆, we obtain that the radial projection of C i (∆) to ∆ coincideswith ∆( i ). Abusing notation slightly, we use C i to also denote the projectiveaction of C i on ∆. With this convention, it is straightforward to check that A ( x ) = C i if and only if x ∈ C i (∆) (note that here we could also replace ∆with the positive cone). Similarly, one has A ( x ) A ( T x ) = C i C j precisely if x ∈ C i (∆) and T x ∈ C j (∆), where T = C − i in this case. That is, we haveequivalence to x ∈ C i (∆) and x ∈ C i C j (∆) ⊂ C i (∆). Inductively, we find that A ( x ) · · · A ( T k x ) = C i · · · C i k if and only if x ∈ C i · · · C i k (∆) . For our case athand, we obtain that x ∈ D if and only if x ∈ M τ ∗ (∆). Note that, as M τ ∗ isprimitive, it acts as a projective contraction on the positive cone. Since eachof C , C is invertible, so is M τ ∗ and the set M τ ∗ (∆) has positive Lebesguemeasure. It follows that the Lebesgue measure (and hence the ν -measure) of D is positive. Finally, to go from α ∈ T to more general α ∈ T , we makeuse of Remark 2.4. (cid:3) Proof of Theorem 1.2
In this section we derive Theorem 1.2 from our work in the previous sections.Let us begin with a discussion of elementary functions on T d and how theyrelate to locally constant functions on ( X, S ), where (
X, S ) is a natural codingof R α associated with the natural measurable partition {F , . . . , F h } . Wedefine η : X → T d by η ( x ) = ω , where ω is the unique point in \ n ∈ N n \ k =0 R − kα ˚ F x k . ERO MEASURE SPECTRUM FOR MULTI-FREQUENCY OPERATORS 13
Let G = \ k ∈ Z R − kα " h [ j =1 ˚ F j , which is a dense G δ set of full Lebesgue measure in T d (by definition of naturalcoding). For ω ∈ G , we can invert this by mapping ω to x = ( x k ) k ∈ Z given by R kα ω ∈ ˚ F x k .Given w = w · · · w n ∈ L ( X ), let F w = n \ k =0 R − kα F w k , which is nonempty by the definition of L ( X ). Let χ w denote the characteristicfunction of F w , and let A denote the algebra generated by { χ w : w ∈ L ( X ) } . Proposition 5.1. If ( X, S ) is a natural coding of R α , then A is ample. Inparticular, A \ { constants } is ample as well.Proof. Given f ∈ C ( T d ) and ε >
0, find δ > | f ( θ ) − f ( θ ) | < ε whenever dist( θ , θ ) < δ . Choose n large enough that for any w ∈ L ( X ) oflength n , diam( F w ) < δ , and define g = X w ∈ L ( X ) | w | = n a w χ w where a w = f ( θ ) for some θ ∈ F w . Clearly g ∈ A and k f − g k ∞ < ε . (cid:3) Proof of Theorem 1.2.
We consider the full measure set of α ’s in T that gen-erate a minimal translation R α : T → T and belong to the full measure setdetermined earlier; compare Proposition 4.1.By these propositions, the minimal translation R α admits a natural codingthat satisfies the Boshernitzan criterion. As R kα is minimal and for any f ∈ A has that its level sets have non-empty interior, the V x are all aperiodic. Thus,by Theorem 2.5, every non-constant locally constant sampling function on thissubshift generates a potential so that the associated Schr¨odinger spectrum isa Cantor set of zero Lebesgue measure.Since the coding is natural, each such locally constant function on the sub-shift corresponds to an elementary function on the torus and the set of func-tions obtained via this correspondence is ample by Proposition 5.1. This con-cludes the proof of the theorem. (cid:3) A Discussion of Possible Extensions to Higher Dimensions
A Road Map to Treating Larger Values of d . Proposition 2.3 is asignificant new result that enabled this project and it is natural to wonder howgeneral it is. The plan for such a result is fairly general.(1) One finds a continued fraction algorithm and obtains S-adic systemsfrom the process applied to a.e. vector in the parameter space.(2) One shows that the resulting shift dynamical systems (a.s.) have purelydiscrete spectrum, and in fact they are measurably isomorphic to a toralrotation and moreover are natural codings thereof.Step (2) requires • An absolutely continuous ergodic invariant measure. • The negativity of the second Lyapanov exponent (of the cocycle thatgives the S-adic system) with respect to the absolutely continuous in-variant measure. • A mild additional assumption. For example either of the following twosuffices. – As in [17, Theorem B] it has a seed point ([17, Definition 64]) andthe second Lyapanov exponent is simple (this is part of the
Pisotcondition [17, Definition 60] in this paper). – As in [8, Theorem 3.1]) it has a periodic Pisot point ([8, Defi-nition 2.4]) with positive range ([8, Definition 2.5]) so that thecorresponding S-adic system (which in this case is a substitutiondynamical system) has discrete spectrum.There are standard approaches to the ergodicity of these algorithms. Forexample, one can relate the continued fraction algorithm to a flow that isknown to be ergodic (see, e.g., [3]) or one can show that it or an accelerationsatisfies some well known conditions (see, e.g., [26, Theorem 8]).The negativity of the second Lyapanov exponent in dimension greater thantwo is shown via computer assisted proof in Hardcastle [20]; see also Berth´e–Steiner–Thuswaldner [7].There is a general strategy [21], but the rigor of these implementationseven in dimension 3 is not always complete [20]. For the Cassaigne-Selmeralgorithm, one can appeal to the 2 dimensional Selmer algorithm (which itis conjugate to) and quote [28] (which appeals to [27] where the result isproven for the closely related Baldwin algorithm) for a proof without computerassistance.6.2.
A Brief Discussion of the Case d = 3 . For translations on T , the4-dimensional Brun algorithm is a natural candidate for the strategy outlined ERO MEASURE SPECTRUM FOR MULTI-FREQUENCY OPERATORS 15 above, and [8, Section 6.4] collects (most of) the necessary inputs. An ana-logue of Proposition 2.3 for the Brun algorithm requires one to verify thatthe second Lyapunov exponent related to the cocycle induced by A on ∆ isnegative. The negativity of the second exponent is unclear to us. In particular[20] experimentally studies this question but is not entirely rigorous. Theother assumptions of [8, Theorem 3.1] are verified in the paragraph before [8,Theorem 6.7]. The result in [8, Theorem 6.7] states the following.
Proposition 6.1.
Let φ be the substitutive realization of the Brun algorithm.For ν B -almost every x ∈ ∆ , the subshift ( X ( φ ( x )) , S ) is a natural coding of ( T , R π ( x ) ) . Given the indeterminate status of Proposition 6.1, we regard the followingproblem as an interesting question for future study.
Conjecture 6.2.
For almost every α ∈ T , the toral translation ( T , R α )admits a natural coding that satisfies Boshernitzan’s criterion.The idea of proof of Conjecture 6.2 relies on Proposition 6.1 and therebyon the question whether the second Lyapunov exponent associated to theBrun algorithm is indeed negative—compare the discussion in the previoussubsection. Assuming Proposition 6.1, we can prove Conjecture 6.2 followingthe same lines as for Proposition 4.1. Here we make use of the observationthat the substitution τ = β ◦ β ◦ β ◦ β is primitive, which can be seenfrom a direct calculation (indeed i ⊳ τ ( j ) for every i and j ); compare thediscussion in [8] preceding Theorem 6.7. With x the right Perron Frobeniuseigenvector of M τ , we have φ ( x ) = τ ∞ . A word builder can be constructed asfollows. If τ [ n +1 ,n +3] = β ◦ β ◦ β , then a can precede b at level n only if ab ∈ { , , , , , , } . From this we can verify that β ◦ β ◦ β ◦ τ is a word builder, irrespective of its position in a sequence ( τ n ) n ∈ N . References [1] P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals,
Bull. Belg. Math. Soc. SimonStevin (2001), 181–207.[2] P. Arnoux, S. Labb´e, On some symmetric multidimensional continued fraction algo-rithms, Ergodic Theory Dynam. Systems (2018), 1601–1626.[3] P. Arnoux, A. Nogueira, Mesures de Gauss pour des algorithmes de fractions continuesmultidimensionnelles (French), Ann. Sci. ´Ecole Norm. Sup. (1993), 645–664.[4] A. Avila, J. Bochi, D. Damanik, Cantor spectrum for Schr¨odinger operators with po-tentials arising from generalized skew-shifts, Duke Math. J. (2009), 253–280. “Note that I use the term “proof” here, despite the fact that I do not attempt to controlround-off errors. I will leave the issue of whether the term “proof” is appropriate to theindividual reader.” [20, Page 132 bottom of left hand side]. [5] A. Avila, J. Bochi, D. Damanik, Opening gaps in the spectrum of strictly ergodicSchr¨odinger operators, J. Eur. Math. Soc. (2012), 61–106.[6] A. Avila, D. Damanik, Z. Zhang, Singular density of states measure for subshift andquasi-periodic Schr¨odinger operators, Commun. Math. Phys. (2014), 469–498.[7] V. Berth´e, W. Steiner, J. M. Thuswaldner, On the second Lyapunov exponent ofsome multidimensional continued fraction algorithms,
Math. of Comp. , to appear.arXiv:1910.09386.[8] V. Berth´e, W. Steiner, J. M. Thuswaldner, Multidimensional continued fractions andsymbolic codings of toral translations, preprint (arXiv:2005.13038).[9] M. Boshernitzan, A condition for unique ergodicity of minimal symbolic flows,
ErgodicTheory Dynam. Systems (1992), 425–428.[10] J. Cassaigne, S. Labb´e, J. Leroy, A set of sequences of complexity 2n + 1, in Com-binatorics on Words , Lecture Notes in Comput. Sci. , Springer, Cham, 2017,144–156.[11] N. Chevallier, Coding of a translation of the two-dimensional torus,
Monatsh. Math. (2009), 101–130.[12] D. Damanik, Schr¨odinger operators with dynamically defined potentials,
Ergodic The-ory Dynam. Systems (2017), 1681–1764.[13] D. Damanik, D. Lenz, A condition of Boshernitzan and uniform convergence in theMultiplicative Ergodic Theorem, Duke Math. J. (2006), 95–123.[14] D. Damanik, D. Lenz, Zero-measure Cantor spectrum for Schr¨odinger operators withlow-complexity potentials,
J. Math. Pures Appl. (9) (2006), 671–686.[15] F. Durand, Linearly recurrent subshifts have a finite number of non-periodic subshiftfactors, Ergodic Theory Dynam. Systems (2000), 1061–1078.[16] L. H. Eliasson, Floquet solutions for the 1-dimensional quasi-periodic Schr¨odinger equa-tion, Commun. Math. Phys. (1992), 447–482.[17] N. P. Fogg, C. Noˆus, Symbolic coding of linear complexity for generic translations ofthe torus, using continued fractions, preprint (arXiv:2005.12229)[18] M. Goldstein, W. Schlag, On resonances and the formation of gaps in the spectrum ofquasi-periodic Schr¨odinger equations,
Ann. of Math. (2011), 337–475.[19] M. Goldstein, W. Schlag, M. Voda, On the spectrum of multi-frequency quasiperiodicSchr¨odinger operators with large coupling,
Invent. Math. (2019), 603–701.[20] D. M. Hardcastle, The three-dimensional Gauss algorithm is strongly convergent almosteverywhere.
Experiment. Math. (2002), no. 1, 131–141.[21] D. M. Hardcastle, K. Khanin, On almost everywhere strong convergence of multi-dimensional continued fraction algorithms. Ergodic Theory Dynam. Systems (2000),1711–1733.[22] Y. Last, M. Shamis, Zero Hausdorff dimension spectrum for the almost Mathieu oper-ator, Commun. Math. Phys. (2016), 729–750.[23] C. Marx, S. Jitomirskaya, Dynamics and spectral theory of quasi-periodic Schr¨odinger-type operators,
Ergodic Theory Dynam. Systems (2017), 2353–2393.[24] J. Puig, Cantor spectrum for the almost Mathieu operator, Commun. Math. Phys. (2004), 297–309.[25] G. Rauzy, Nombres alg´ebriques et substitutions,
Bull. Soc. Math. France (1982),147–178.[26] F. Schweiger,
Multidimensional Continued Fractions , Oxford University Press, Oxford,2000.
ERO MEASURE SPECTRUM FOR MULTI-FREQUENCY OPERATORS 17 [27] F. Schweiger, Invariant measure and exponent of convergence for Baldwin’s algorithmGCFP, ¨Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II (2002), 11–23.[28] F. Schweiger, Ergodic and Diophantine properties of algorithms of Selmer type,
ActaArith. (2004), 99–111.[29] C. Steineder, R. Winkler, Complexity of Hartman sequences,
J. Th´eor. Nombres Bor-deaux (2005), 347–357. Department of Mathematics, University of Utah, Salt Lake City, UT 84112,USA
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