Sums of regular Cantor sets of large dimension and the Square Fibonacci Hamiltonian

0.44 MB·

by  David Damanik

SUMS OF REGULAR CANTOR SETS OF LARGE DIMENSION
AND THE SQUARE FIBONACCI HAMILTONIAN
DAVIDDAMANIKANDANTONGORODETSKI
6
1
Abstract. Weshowthatundernaturaltechnicalconditions,thesumofaC2
0
dynamically defined Cantor set with a compact set in most cases (for almost
2
all parameters) has positive Lebesgue measure, provided that the sum of the

n Hausdorff dimensions of these sets exceeds one. As an application, we show
a that for many parameters, the Square Fibonacci Hamiltonian has spectrum
J of positive Lebesgue measure, while at the same time the density of states

7 measureispurelysingular.

]
S
D
1. Introduction
.
h
1.1. Sums of dynamically defined Cantor sets. The study of the structure
t
a andthepropertiesofsumsofCantorsetsismotivatedbyapplicationsindynamical
m
systems [37, 38, 39, 40], number theory [7, 24, 32], harmonic analysis [3, 4], and
[ spectral theory [14, 15, 16, 56]. In many cases dynamically defined Cantor sets are

of special interest.
1
v
Definition 1. A dynamically defined (or regular) Cantor set of class Cr is a Can-
9
tor subset C ⊂ R of the real line such that there are disjoint compact intervals
3
6 I1,...,Il ⊂R and an expanding Ck function Φ:I1∪···∪Il →I from the disjoint
1 union I ∪···∪I to its convex hull I with
1 l
0 (cid:92)
. C = Φ−n(I).
1
0 n∈N
6 In the case when the restriction of the map Φ to each of the intervals I ,j =
j
1 1,...,l,isaffine,thecorrespondingCantorsetisalsocalledaffine. Ifalltheseaffine
:
v maps have the same expansion rate (i.e., |Φ(cid:48)(x)| = const for all x ∈ I ∪···∪I ),
1 l
i theCantorsetiscalledhomogeneous. AspecificexampleofahomogeneousCantor
X
set, a middle-α Cantor set C , is defined by Φ : [0,a]∪[1−a,1] → [0,1], where
r a
a Φ(x)= xa for x∈[0,a], and Φ(x)= xa − a1 +1 for x∈[1−a,1]. For example, C1/3
is the standard middle-third Cantor set.
Considering the sum C+C(cid:48) of two Cantor sets C,C(cid:48), defined by
C+C(cid:48) ={c+c(cid:48) :c∈C, c(cid:48) ∈C(cid:48)},
itisnothardtoshow(see, e.g., Chapter4in[40])thatiftheCantorsetsC andC(cid:48)
aredynamicallydefined,onehasdim (C+C(cid:48))≤min(dim C+dim C(cid:48),1). Hence
H H H
in the case dim C +dim C(cid:48) < 1, the sum C +C(cid:48) must be a Cantor set, and an
H H
interestingquestionhereiswhethertheidentitydim (C+C(cid:48))=dim C+dim C(cid:48)
H H H
Date:January8,2016.
D.D.wassupportedinpartbyNSFgrantsDMS–1067988andDMS–1361625.
A.G.wassupportedinpartbyNSFgrantDMS–1301515.
1
2 D.DAMANIKANDA.GORODETSKI
holds. This question was addressed for homogeneous Cantor sets in [44] (see also
[35]), and some explicit criteria were provided in [21].
Inthecasewhendim C+dim C(cid:48) >1,amajorresultwasobtainedbyMoreira
H H
and Yoccoz in [34]. They showed that for a generic pair of Cantor sets (C,C(cid:48)) in
thisregime, thesumC+C(cid:48) containsaninterval. Thegenericityassumptionsthere
arequitenon-explicit,andcannotbeverifiedinaspecificcase. Thisdoesnotallow
one to apply this result when a specific pair or a specific family of Cantor sets is
given (which is often the case in applications), which therefore motivates further
investigations in this direction. For example, while [34] solves one part of the Palis
conjectureonsumsofCantorsets(“genericallythesumoftwodynamicallydefined
Cantorsetseitherhaszeromeasureorcontainsaninterval”),thesecondpartofthe
conjecture (“generically the sum of two affine Cantor sets either has zero measure
or contains an interval”) is still open.
An important characteristic of a Cantor set related to questions about inter-
sections and sum sets is the thickness, usually denoted by τ(C). This notion was
introduced by Newhouse in [37]; for a detailed discussion, see [40]. The famous
Newhouse Gap Lemma asserts that if τ(C)·τ(C(cid:48)) > 1, then C +C(cid:48) contains an
interval. This allowed for essential progress in dynamics [38, 39, 13], and found
an application in number theory [1]. Nevertheless, in some cases τ(C)·τ(C(cid:48))<1,
while dim C+dim C(cid:48) >1, and other arguments are needed.
H H
In [53] Solomyak studied the sums C +C of middle-α type Cantor sets. He
a b
showed that in the regime when dim C +dim C(cid:48) > 1, for almost every pair of
H H
parameters (a,b), one has Leb(C +C ) > 0. Similar results for sums of homoge-
a b
neous Cantor sets (parameterized by the expansion rate) with a fixed compact set
were obtained in [44].
In this paper we are able to work in far greater generality, and our first main
result reads as follows:
Theorem 1.1. Let {C } be a family of dynamically defined Cantor sets of class
λ
C2 (i.e., C =C(Φ ), where Φ is an expansion of class C2 both in x∈R and in
λ λ λ
λ ∈ J = (λ ,λ )) such that d dim C (cid:54)= 0 for λ ∈ J. Let K ⊂ R be a compact
0 1 dλ H λ
set such that
dim C +dim K >1 for all λ∈J.
H λ H
Then Leb(C +K)>0 for a.e. λ∈J.
λ
Remark 1.2. ItwouldbeinterestingtorelaxtheassumptionsinTheorem1.1and
to show that the same statement holds for C1+α Cantor sets. We conjecture that
this is indeed the case (possibly under some extra conditions on the dependence of
Φ and ∂ Φ on λ).
∂x
In the case when the dynamically defined Cantor sets {C } are affine (or non-
λ
linear, but C2-close to affine), a statement analogous to Theorem 1.1 was obtained
in [20]. The case of a sum of homogeneous (affine with the same contraction rate
for each of the generators) Cantor sets with a dynamically defined Cantor set was
considered in Theorem 1.4 in [51]; in this case the set of exceptional parameters
has zero Hausdorff dimension.
In order to put these results in perspective, we summarize them in the following
table. (Below we always set d =dim C (or d =dim C ) and d =dim K. In
1 H 1 H λ 2 H
the case d +d <1 we ask whether it is true that dim (C+K)=d +d , and in
1 2 H 1 2
SUMS OF CANTOR SETS AND THE SQUARE FIBONACCI HAMILTONIAN 3
the case d +d >1 we ask whether it is true that C+K contains an interval, and
1 2
if this is unknown, whether Leb(C+K)>0.)
What is known about sums of dynamically defined Cantor sets:
d +d <1 d +d >1
1 2 1 2
ForaCrgenericpairofCan-
Yes, follows from [21]. C+K contains an interval [34]
tor sets (C,K)
Given a family {C }
λ λ∈J Yes, for a.e. parameter, fol- Leb(C +K)>0 for a.e. λ,
such that d (dim C )(cid:54)=0, λ
dλ H λ lows from [21]. this paper
and a compact K ⊂R
Leb(C + K) > 0 [20];
For a generic pair of affine whether C +K contains an
Yes, follows from [21].
Cantor sets (C,K) interval is an open part of
the Palis conjecture
Leb(C +K) > 0 for a.e. λ
Given a family {C } of λ
λ λ∈J [44] (see also [51]); whether
homogeneous self-similar
Yes, for a.e. parameter, [44] C + K contains an inter-
Cantor sets and a compact λ
val for a.e. parameter is un-
K ⊂R
known
Leb(C + K) > 0, for a.e.
Givenafamily{C } a
a a∈(0,1/2) parameter a, [53]; whether
of middle-α Cantor sets, Yes, with countable number
C + K contains an inter-
take K = C for some b ∈ of exceptions, [43] a
b
val for a.e. parameter is un-
(0,1/2)
known.
There exists an example
of two dynamically defined
Cantor sets such that C +
For a specific fixed pair of
Explicitconditionsaregiven K is a Cantor set of posi-
dynamically defined Cantor
in [21] tivemeasure,[50]. Noverifi-
sets (C,K)
ablecriteriaareknownwhen
thickness arguments do not
help.
One should also mention the results [23, 28, 42, 52] in the spirit of Marstrand’s
theorem on properties of sum sets of the form C +λC (which is equivalent to the
1 2
projection of the product C ×C along the line of slope λ).
1 2
In many applications a dynamically defined Cantor sets appears as the inter-
section of the stable lamination of some hyperbolic horseshoe with a transversal.
4 D.DAMANIKANDA.GORODETSKI
More specifically, suppose that f : M2 → M2 is a Cr-diffeomorphism, r ≥ 2, and
Λ ⊂ M2 is a hyperbolic horseshoe (i.e., a totally disconnected locally maximal in-
variant compact set such that there exists an invariant splitting T = Es⊕Eu so
Λ
that along the stable subbundle {Es}, the differential Df contracts uniformly, and
along {Eu}, the differential of the inverse Df−1 contracts uniformly). Then
Ws(Λ)={x∈M2 :dist(fn(x),Λ)→0 as n→+∞}
consists of stable manifolds Ws(Λ) = (cid:83) Ws(x) and locally looks like a prod-
x∈Λ
uct of a Cantor set with an interval. If f = fλ∗ ∈ {fλ}λ∈J=(λ0,λ1) is an element
of a smooth family of diffeomorphisms, then there exists a family of horseshoes
{Λ },f (Λ ) = Λ , for parameters λ sufficiently close to the initial λ∗ ∈ J. Sup-
λ λ λ λ
posethatL⊂M2isalinetransversaltoeveryleafinWs(Λ ),λ∈J,withcompact
λ
intersection L∩Ws(Λ ). The intersection C =L∩Ws(Λ ) is a λ-dependent dy-
λ λ λ
namicallydefinedCantorset. Thelamination{Ws(x)}consistsofCr leaves,butin
generalonecannotincludeitinafoliationofsmoothnessbetterthanC1+α(evenfor
C∞ orrealanalyticf). ThatjustifiesthetraditionalassumptiononC1+α smooth-
ness of generators of a dynamically defined Cantor set1. This prevents us from
using Theorem 1.1 in the context above. Nevertheless, the analog of Theorem 1.1
holds for families of Cantor sets {C } obtained via the described construction:
λ
Theorem1.3. Supposethat{f } ,f :M2 →M2,isaC2-familyofC2-
λ λ∈J=(λ0,λ1) λ
diffeomorphismswithuniformly(inλ)boundedC2 norms. Let{Λ } beafamily
λ λ∈J
of hyperbolic horseshoes, and {L } be a smooth family of curves parameterized
λ λ∈J
by γ : R → M2, transversal to Ws(Λ ), with compact C = γ−1(L ∩Ws(Λ )).
λ λ λ λ λ λ
Assume that
d
(1) dim C (cid:54)=0 for all λ∈J.
H λ
dλ
If K ⊂R is a compact set such that
(2) dim C +dim K >1 for all λ∈J,
H λ H
then Leb(C +K)>0 for Lebesgue almost every λ∈J.
λ
NoticethatTheorem1.3impliesTheorem1.1. Indeed,undertheassumptionsof
Theorem1.1onecanconstructafamilyofhorseshoesandcurvesasinTheorem1.3
that produce the same family of Cantor sets {C }. We prove Theorem 1.3 in
λ
Section 2.
1.2. An Application to the Square Fibonacci Hamiltonian. The square Fi-
bonacci Hamiltonian is the bounded self-adjoint operator
[H(2) ψ](m,n)=ψ(m+1,n)+ψ(m−1,n)+ψ(m,n+1)+ψ(m,n−1)+
λ1,λ2,ω1,ω2
(cid:16) (cid:17)
+ λ χ (mα+ω mod 1)+λ χ (nα+ω mod 1) ψ(m,n)
1 [1−α,1) 1 2 [1−α,1) 2
√
in (cid:96)2(Z2), with α = 5−1, coupling constants λ ,λ > 0 and phases ω ,ω ∈ T =
2 1 2 1 2
R/Z. The standard Fibonacci Hamiltonian is the bounded self-adjoint operator
[H(1)ψ](n)=ψ(n+1)+ψ(n−1)+λ χ (nα+ω mod 1)ψ(n)
λ,ω 1 [1−α,1) 1
1Notice that C1-smoothness is usually too weak since it does not allow one to use distortion
propertyarguments;see[33,55]forsomeresultsonsumsofC1 Cantorsets.
SUMS OF CANTOR SETS AND THE SQUARE FIBONACCI HAMILTONIAN 5
in(cid:96)2(Z),againwiththecouplingconstantλ>0andthephaseω ∈T. Forarecent
surveyofthespectraltheoryoftheFibonacciHamilonianandthesquareFibonacci
Hamiltonian, see [8].
Using the minimality of an irrational rotation and strong operator convergence,
onecanreadilyseethatthespectraoftheseoperatorsarephase-independent. That
is, there are compact subsets Σ and Σ of R such that
λ λ1,λ2
σ(H(1))=Σ for every ω ∈T,
λ,ω λ
σ(H(2) )=Σ for every ω ,ω ∈T.
λ1,λ2,ω1,ω2 λ1,λ2 1 2
Thedensityofstatesmeasuresassociatedwiththeseoperatorfamiliesaredefined
as follows,
(cid:90) (cid:90) (cid:90)
Rg(E)dνλ1,λ2(E)= T T(cid:104)δ0,g(Hλ(21),λ2,ω1,ω2)δ0(cid:105)(cid:96)2(Z2)dω1dω2.
and
(cid:90) (cid:90)
g(E)dνλ(E)= (cid:104)δ0,g(Hλ(1,ω))δ0(cid:105)(cid:96)2(Z)dω.
R T
It is a standard result from the theory of ergodic Schr¨odinger operators that Σ =
λ
suppν and Σ =suppν , where suppν denotes the topological support of
λ λ1,λ2 λ1,λ2
the measure ν.
Thetheoryofseparableoperators(see,e.g.,[11,Appendix]and[48,SectionsII.4
and VIII.10]) quickly implies that
(3) Σ =Σ +Σ
λ1,λ2 λ1 λ2
and
(4) ν =ν ∗ν ,
λ1,λ2 λ1 λ2
where the convolution of measures is defined by
(cid:90) (cid:90) (cid:90)
g(E)d(µ∗ν)(E)= g(E +E )dµ(E )dν(E ).
1 2 1 2
R R R
It was shown in [12] that for every λ > 0, the set Σ is a dynamically defined
λ
Cantor set (see [5, 6, 9] for earlier partial results for sufficiently small or large
values of λ). In particular, its box counting dimension exists, coincides with its
Hausdorff dimension, and the common value belongs to (0,1). As was pointed
out above, a particular consequence of this is that if (λ ,λ ) ∈ R2 is such that
1 2 +
dim Σ +dim Σ < 1, then Σ has zero Lebesgue measure. Here we are
H λ1 H λ2 λ1,λ2
able to prove the following companion result:
Theorem 1.4. Suppose that for all pairs (λ ,λ ) in some open set U ⊂ R2, we
1 2 +
have dim Σ +dim Σ > 1. Then, for Lebesgue almost all pairs (λ ,λ ) ∈ U,
H λ1 H λ2 1 2
Σ has positive Lebesgue measure.
λ1,λ2
Combiningresultsfrom[11]and[12],itfollowsthatforLebesguealmostallpairs
(λ ,λ ) ∈ R2 in the region where dim ν +dim ν > 1, the measure ν is
1 2 + H λ1 H λ2 λ1,λ2
absolutely continuous. We are also able here to prove a companion result for the
latter statement:
Theorem 1.5. Suppose that dim ν +dim ν < 1. Then, ν is singular,
H λ1 H λ2 λ1,λ2
that is, it is supported by a set of zero Lebesgue measure.
6 D.DAMANIKANDA.GORODETSKI
In addition, it was shown in [12] that for every λ>0, we have
(5) 0<dim ν <dim Σ <1.
H λ H λ
This shows in particular that the curves
{(λ ,λ )∈R2 :dim ν +dim ν =1}
1 2 + H λ1 H λ2
and
{(λ ,λ )∈R2 :dim Σ +dim Σ =1}
1 2 + H λ1 H λ2
are disjoint. The complement of the union of these curves consists of three regions,
inwhichwehavethreedifferentkindsofspectralbehaviorduetotheresultsabove
andthediscussionprecedingeachofthem. Wesummarizethesefindingsandmake
the global picture explicit in the following corollary.
Corollary 1.6. Consider the following three regions in R2:
+
U ={(λ ,λ )∈R2 :dim ν +dim ν >1},
acds 1 2 + H λ1 H λ2
U ={(λ ,λ )∈R2 :dim Σ +dim Σ >1 and dim ν +dim ν <1},
pmsd 1 2 + H λ1 H λ2 H λ1 H λ2
U ={(λ ,λ )∈R2 :dim Σ +dim Σ <1}.
zmsp 1 2 + H λ1 H λ2
Then, the following statements hold:
(a) Each of the regions U , U , U is open and non-empty.
acds pmsd zmsp
(b) Theregions U , U , U aredisjoint andthe unionof theirclosures
acds pmsd zmsp
covers the parameter space R2.
+
(c) For Lebesgue almost every (λ ,λ )∈U , ν is absolutely continuous,
1 2 acds λ1,λ2
and hence Σ has positive Lebesgue measure.
λ1,λ2
(d) Forevery(λ ,λ )∈U ,ν issingular,butforLebesguealmostevery
1 2 pmsd λ1,λ2
(λ ,λ )∈U , Σ has positive Lebesgue measure.
1 2 pmsd λ1,λ2
(e) For every (λ ,λ ) ∈ U , Σ has zero Lebesgue measure, and hence
1 2 zmsp λ1,λ2
ν is singular.
λ1,λ2
Remark1.7. (a)ThepotentialoftheFibonacciHamiltonianmaybegeneratedby
the Fibonacci substitution a(cid:55)→ab, b(cid:55)→a. This substitution is the most prominent
example of an invertible two-letter substitution. We believe that, using [19, 30],
the results above may be generalized to the case where the Fibonacci substitution
is replaced by a general invertible two-letter substitution.
(b)Weexpectthatsimilarphenomenacanappearalsoinothermodels, suchasfor
example the labyrinth model, or the square off-diagonal (or tridiagonal) Fibonacci
Hamiltonian.
Thecoexistenceofpositivemeasurespectrumandsingulardensityofstatesmea-
sure is a rather unusual phenomenon. Until very recently it was an open problem
whetherthiscanevenoccurinthecontextofSchr¨odingeroperators. Theexistence
of Schr¨odinger operators with quasi-periodic potentials exhibiting this phenome-
non was shown in [2]. However, the examples given in that paper are somewhat
artificial, and “typical” quasi-periodic Schr¨odinger operators are not expected to
have these two properties. The examples provided by the Square Fibonacci Hamil-
tonian with parameters in U , on the other hand, are not artificial at all, but
pmsd
rathercorrespondtooperatorsthatarearguablyphysicallyrelevant. Moreoverthe
phenomenon is made possible by and is closely connected to the strict inequality
between dim ν and dim Σ , as stated in (5), which was originally conjectured
H λ H λ
SUMS OF CANTOR SETS AND THE SQUARE FIBONACCI HAMILTONIAN 7
by Barry Simon and finally proved in [12] (see [10] for an earlier partial result for
sufficiently small values of λ).
2. Sums of Dynamically Defined Cantor Sets
In this section we prove Theorem 1.3. The proof is based on Theorem 3.7 from
[11]. The setting there is the following.
Suppose J ⊂ R is a compact interval, and f : M2 → M2, λ ∈ J, is a smooth
λ
family of smooth surface diffeomorphisms. Specifically, we require f (p) to be C2-
λ
smooth with respect to both λ and p, with a finite C2-norm. Also, we assume
that f : M2 → M2, λ ∈ J, has a locally maximal transitive totally disconnected
λ
hyperbolic set Λ that depends continuously on the parameter.
λ
Let γ :R→M2 be a family of smooth curves, smoothly depending on the pa-
λ
rameter, and L =γ (R). Suppose that the stable manifolds of Λ are transversal
λ λ λ
to L .
λ
Lemma 2.1 (Lemma 3.1 from [11]). There is a Markov partition of Λ and a
λ
continuous family of projections π : Λ → L along stable manifolds of Λ such
λ λ λ λ
that for any two distinct elements of the Markov partition, their images under π
λ
are disjoint.
Suppose σ :Σ(cid:96) →Σ(cid:96) is a topological Markov chain, which for every λ∈J is
A A A
conjugated to f :Λ →Λ via the conjugacy H :Σ(cid:96) →Λ . Let µ be an ergodic
λ λ λ λ A λ
probability measure for σ : Σ(cid:96) → Σ(cid:96) such that h (σ ) > 0. Set µ = H (µ),
A A A µ A λ λ
then µ is an ergodic invariant measure for f :Λ →Λ .
λ λ λ λ
Let π : Λ → L be the continuous family of continuous projections along
λ λ λ
the stable manifolds of Λ provided by Lemma 2.1. Set ν = γ−1 ◦ π (µ ) =
λ λ λ λ λ
γ−1◦π ◦H (µ).
λ λ λ
In this setting the following theorem holds.
Theorem 2.2 (Theorem 3.7 from [11]). Suppose that J is a compact interval so
that (cid:12)(cid:12)ddλLyapu(µλ)(cid:12)(cid:12)≥δ >0 for some δ >0 and all λ∈J. Then for any compactly
supported exact-dimensional measure η on R with
dim η+dim ν >1
H H λ
forallλ∈J, theconvolutionη∗ν isabsolutelycontinuouswithrespecttoLebesgue
λ
measure for Lebesgue almost every λ∈J.
Remark 2.3. Infact, inTheorem2.2theconditiononexactdimensionalityofthe
measure η can be replaced by the following one (and this is the only consequence
of exact dimensionality of η that was used in the proof of Theorem 2.2 in [11]):
• There are C > 0 and d > 0 such that for every x ∈ R and r > 0, we have
η(B (x))≤Crd.
r
Proof of Theorem 1.3. The condition dim C +dim K >1 trivially implies that
H λ H
dim K > 0. By Frostman’s Lemma (see, e.g., [27, Theorem 8.8]), for every d <
H
dim K, there exist a Borel measure η on R with η(K)=1 and a constant C such
H
that
(6) η(B (x))≤Crd for every x∈R and r >0.
r
We will show that for every λ ∈ J, there exists ε = ε(λ ) > 0 such that
0 0
Leb(C +K) > 0 for Lebesgue almost every λ ∈ (λ −ε,λ +ε)∩J. This will
λ 0 0
imply Theorem 1.3.
8 D.DAMANIKANDA.GORODETSKI
Fix λ ∈J. Let µ be the equilibrium measure on Λ that corresponds to the
0 λ0 λ0
potential −dimHCλ0log|Dfλ0|Eu|. Then (see [29]), the measure µλ0 is a measure
of maximal (unstable) dimension, that is, dim π (µ ) = dim C . Denote by
H λ0 λ0 H λ0
ν the projection π (µ ). In order to mimic the setting of Theorem 2.2, set µ=
λ0 λ0 λ0
H−1(µ ). Then µ is an invariant probability measure for the shift σ :Σl →Σl .
λ0 λ0 A A A
Let us denote µ = H (µ) and ν = π (µ ). There exists a canonical family of
λ λ λ λ λ
conjugaciesH :Λ →Λ ,(λ ,λ )∈J×J,sothatH ◦f =f ◦H .
λ1,λ2 λ1 λ2 1 2 λ1,λ2 λ1 λ2 λ1,λ2
It is well known (see, for example, Theorem 19.1.2 from [22]) that each of the
maps H is H¨older continuous. Moreover, the H¨older exponent tends to one as
λ1,λ2
|λ −λ |→0; see [41]. As a result, we conclude that for any λ sufficiently close to
1 2
λ , we have
0
dim ν +d>1
H λ
for a suitable d that is chosen sufficiently close to dim K and for which we have
H
(6) with suitable η and C.
In order to apply Theorem 2.2 we need to show that (cid:12)(cid:12)ddλLyapu(µλ)(cid:12)(cid:12) ≥ δ > 0.
But due to [26] we know that
h
Lyapuµ = µλ ,
λ
dim ν
H λ
where h =h is the entropy of the invariant measure µ (which is by construc-
µλ0 µ λ
tion independent of λ). Notice also that Lyapuµ is a C1 smooth function of λ.
λ
Indeed, the center-stable and center-unstable manifolds of the partially hyperbolic
invariant set of the map (λ,p)(cid:55)→(λ,f (p)) are C2-smooth, hence
λ
(cid:90) (cid:90)
Lyapuµλ = log|Dfλ|Eu|dµλ = log|Dfλ(Hλ(ω))|Eu|dµ(ω)
Λλ ΣlA
is a C1-smooth function of λ∈J.
Finally, consider dim C and dim ν as functions of λ; see Fig. 2. Due to [25]
H λ H λ
we know that dim C is a C1-function of λ. Without loss of generality we can
H λ
assume that d dim C ≥ δ > 0 for some δ > 0. Since suppν ⊆ C , we have
dλ H λ λ λ
dim ν ≤ dim C . By construction we have dim ν = dim C . This implies
H λ H λ H λ0 H λ0
that d | dim ν = d | dim C ≥ δ > 0, and hence for some ε > 0,
dλ λ=λ0 H λ dλ λ=λ0 H λ
d dim ν ≥ δ >0 for λ∈(λ −ε,λ +ε). Now we can apply Theorem 2.2 to the
dλ H λ 2 0 0
measuresη andν , andgetthatforLebesguealmosteveryλ∈(λ −ε,λ +ε), the
λ 0 0
convolution η∗ν is absolutely continuous with respect to Lebesgue measure, and
λ
hence Leb(C +K)>0. (cid:3)
λ
3. The Square Fibonacci Hamiltonian
3.1. A Dynamical Description of the Spectrum of the Fibonacci Hamil-
tonian. There is a fundamental connection between the spectral properties of the
Fibonacci Hamiltonian and the dynamics of the trace map
(7) T :R3 →R3, T(x,y,z)=(2xy−z,x,y).
SUMS OF CANTOR SETS AND THE SQUARE FIBONACCI HAMILTONIAN 9
dimHCλ
dimHνλ
λ0 λ
Figure 1. Graphs of dim C and dim ν as functions of λ
H λ H λ
The function G(x,y,z)=x2+y2+z2−2xyz−1 is invariant2 under the action of
T, and hence T preserves the family of cubic surfaces3
(cid:26) λ2(cid:27)
(8) S = (x,y,z)∈R3 :x2+y2+z2−2xyz =1+ .
λ
4
It is therefore natural to consider the restriction T of the trace map T to the
λ
invariant surface S . That is, T : S → S , T = T| . We denote by Λ the set
λ λ λ λ λ Sλ λ
of points in S whose full orbits under T are bounded (it follows from [5, 49] that
λ λ
Λ is equal to the non-wandering set of T ; compare the discussion in [10]).
λ λ
Denote by (cid:96) the line
λ
(cid:26)(cid:18) (cid:19) (cid:27)
E−λ E
(9) (cid:96) = , ,1 :E ∈R .
λ
2 2
It is easy to check that (cid:96) ⊂ S . The key to the fundamental connection between
λ λ
thespectralpropertiesoftheFibonacciHamiltonianandthedynamicsofthetrace
map is the following result of Su¨t˝o [54]. An energy E ∈R belongs to the spectrum
Σ of the Fibonacci Hamiltonian if and only if the positive semiorbit of the point
λ
(E−λ,E,1) under iterates of the trace map T is bounded.
2 2
It turns out that for every λ > 0, Λ is a locally maximal compact transitive
λ
hyperbolic set of T : S → S ; see [5, 6, 9]. Moreover, it was shown in [12] that
λ λ λ
for every λ > 0, the line of initial conditions (cid:96) intersects Ws(Λ ) transversally.
λ λ
Thus, we are essentially in the setting in which Theorem 1.3 applies. The only
minor difference is that in the present setting, the surface S depends formally on
λ
λ, while it is λ-independent in the setting of Theorem 1.3. After partitioning the
parameter space into smaller intervals if necessary, we can then consider a small
2GisusuallycalledtheFricke-Vogtinvariant.
3ThesurfaceS0 isknownasCayleycubic.
10 D.DAMANIKANDA.GORODETSKI
λ-interval, choose a λ in it, and then conjugate with smooth projections of S to
0 λ
S .
λ0
3.2. Proof of Theorem 1.4. Using the connection between the spectrum of the
(one-dimensional) Fibonacci Hamiltonian and the dynamics of the trace map, we
can now derive Theorem 1.4 from Theorem 1.3.
Proof of Theorem 1.4. It clearly suffices to work locally in U. That is, we consider
a rectangular box B = {(λ ,λ ) : a < λ < b, c < λ < d} inside U and prove
1 2 1 2
thatforLebesguealmostevery(λ ,λ )∈B,Σ haspositiveLebesguemeasure.
1 2 λ1,λ2
To accomplish this, it suffices to show that for every fixed λ ∈ (c,d), Σ has
2 λ1,λ2
positive Lebesgue measure for Lebesgue almost every λ ∈(a,b).
1
The set Σ will play the role of the set K in Theorem 1.3. By the analyticity
λ2
of λ (cid:55)→dim Σ , we can subdivide (a,b) into intervals, on the interiors of which
1 H λ1
we have the condition
d
dim Σ (cid:54)=0.
dλ H λ1
1
Thisensuresthatcondition(1)inTheorem1.3holds. Condition(2)inTheorem1.3
holds since we work inside U. All the other assumptions in Theorem 1.3 hold by
the discussion in the previous subsection. Thus we may apply Theorem 1.3 and
obtain the desired statement. (cid:3)
3.3. Proof of Theorem 1.5. Let us begin by recalling some basic concepts from
measure theory and fractal geometry; the standard texts [17, 27] can be consulted
for background information. Suppose µ is a finite Borel measure on Rd. The lower
Hausdorff dimension, resp. the upper Hausdorff dimension, of µ are given by
(10) dim−(µ)=inf{dim (S):µ(S)>0},
H H
(11) dim+(µ)=inf{dim (S):µ(Rd\S)=0}.
H H
These dimensions can be interpreted in the following way. The measure µ gives
zero weight to every set S with dim (S) < dim−(µ) and, for every ε > 0, there is
H H
a set S with dim (S)<dim+(µ)+ε that supports µ (i.e., µ(R\S)=0).
H H
For x ∈ Rd and ε > 0, we denote the open ball with radius ε and center x by
B(x,ε). The lower scaling exponent of µ at x is given by
logµ(B(x,ε))
α−(x)=liminf .
µ ε→0 logε
For µ-almost every x, α−(x)∈[0,d]. Moreover, we have
(12) dim−(µ)=µ−essinfα− ≡sup{α:α−(x)≥α for µ-almost every x},
H µ µ
(13) dim+(µ)=µ−esssupα− ≡inf{α:α−(x)≤α for µ-almost every x},
H µ µ
compare [18, Propositions 10.2 and 10.3].
One can also consider the upper scaling exponent of µ at x,
logµ(B(x,ε))
α+(x)=limsup ,
µ logε
ε→0
which also belongs to [0,d] for µ-almost every x. The measure µ is called exact-
dimensional if there is a number dimµ ∈ [0,d] such that α+(x) = α−(x) = dimµ
µ µ
for µ-almost every x ∈ Rd. In this case, it of course follows that dim+(µ) =
H
dim−(µ) = dimµ, and tangentially we note that the common value also coincides
H