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Preface: a description of contents

Decay of various quantities (return or survival probability, correlation functions)

in time are the basis of a multitude of important and interesting phenomena in

quantum physics, ranging from spectral properties, resonances, return and approach

to equilibrium, to dynamical stability properties and irreversibility and the “arrow of

time”. This monograph is devoted to a clear and precise, yet (hopefully) pedagogical

account of the associated concepts and methods. It is aimed at graduate students

and researchers in the fields of mathematical physics and mathematics.

v


vi Asymptotic Decay in Quantum Physics


Contents

Preface: a description of contents v

Appendix A A survey of classical ergodic theory 1

Appendix B Transfer matrix, Prufer variables and spectral analysis

of sparse models 23

Appendix C Symmetric Cantor sets and related subjects 41

Bibliography 49

vii


Appendix A

A survey of classical ergodic theory

In this appendix we discuss some of the elements of classical ergodic theory, together

with some rudiments of number theory, which are complementary to the discussion

in chapters ?? and ??. For complete and readable expositions, see [36], [54] and [7],

and for a discussion in the spirit of this appendix, which emphasizes the relationship

with number theory and probability, see [25].

The ergodic problems of classical mechanics have been covered in various clas-

sical monographs, notably [2] and [50]. A lucid treatment of statistical mechanics

aspects may be found in [50] and the still pedagogically valuable [34].

To Francois Viete (1540-1603), who is by some considered as the “father of

algebra” (see the fascinating history in the article by J. J. O’Connor and E. F.

Robertson in Wikipedia) is attributed the formula (Vieta’s formula):

sinx

x=

∞∏k=1

cos(2−kx) (A.1)

This formula is used twice in this book: in the beginning of chapter ?? as the

basis of the most elementary example that the convolution of two s.c. measures

may be a.c., and in the approach to equilibrium of the spin model in chapter ??.

It is therefore quite adequate to use it, following [25], in order to provide a first

illustration of the connection between number theory, analysis and probability.

Given a real number x ∈ [0, 1), and an integer q ≥ 2, there exists one and only

one representation of x in the basis q given by the expansion

x =

∞∑k=1

εkqk

=ε1q

+ε2q2

+ · · ·+ εkqk

+ · · · (A.2)

where εk are integers such that

0 ≤ εk < q for k = 1, 2, . . . (A.3)

and

εk < q − 1 for any infinity of k’s (A.4)

If q = 10, (A.2) coincides with the decimal representation

x = 0.ε1ε2 · · · εk · · · (A.5)

1


2 Asymptotic Decay in Quantum Physics

We shall use the same notation to represent x in basis q, with q 6= 10. Except when

explicitly mentioned, we shall use q = 2. In this case, (A.2), (A.3,b) establishes a

1-1 correspondence between a real number x ∈ [0, 1] and a binary sequence

ω = (εj)j≥1 with εj ∈ 0, 1 (A.6)

with an infinity of digits εj equal to zero: note that the latter ((A.4) in general)

chooses one among the two possible representations, e.g.,

3/4 = 1/2 + 1/22 + 0/23 + 0/24 + · · · = 1/2 + 0/22 + 1/23 + 1/24 + · · ·

Note also that in (A.6), εjj≥1 varies with x, and we denote this by

x⇔ εj(x)j≥1 (A.7)

For example,

ε1(x) =

0 if x ∈ [0, 1/2) ,

1 if x ∈ [1/2, 1) ,(A.8)

ε2(x) =

0 if x ∈ [0, 1/4) ∪ [1/2, 3/4) ,

1 if x ∈ [1/4, 1/2) ∪ [3/4, 1) ,(A.9)

etc. We have

Definition A.1. For each integer k, the k–th Rademacher function is given by

rk(x) = 1− 2εk(x) (A.10)

For example, from (A.8,b)

r1(x) =

1 if x ∈ [0, 1/2) ,

−1 if x ∈ [1/2, 1) ,(A.11)

r2(x) =

1 if x ∈ [0, 1/4) ∪ [1/2, 3/4) ,

−1 if x ∈ [1/4, 1/2) ∪ [3/4, 1) ,(A.12)

etc. Whatever n, rn(x) is constant and equal to ±1 in the intervals

In,j ≡ [j/2n, (j + 1)/2n) (A.13)

with j = 0, 1, . . . , 2n − 1, alternating sign when x varies from one interval to the

next, with rn(0) = 1.

(A.10) may thus be expressed as

rj(x) = sign(sin 2jπx) (A.14)

where

sign(y) ≡

1 if y > 0 ,

0 if y = 0 ,

−1 if y < 0 ,

. (A.15)


A survey of classical ergodic theory 3

From the definition or (A.14), rj(x) changes sign an odd number of times in each

interval Ii,k = [k/2i, (k + 1)/2i) of continuity of ri(x) if i < j (look at (A.11,b)). It

follows that ∫ 1

0

ri(x)rj(x)dx = δij (A.16)

The Rademacher functions do not form a complete orthonormal basis of L2(0, 1),

because they may be expressed as a particular linear combination of the so-called

Haar functions, which do form a complete orthonormal basis, see [26].

We now use (A.2) with q = 2 and (A.10) to write

1− 2x = 1− 2

∞∑k=1

εk/2k =

∞∑k=1

1− 2εk2k

=

∞∑k=1

rk(x)

2k

On the other hand,∫ 1

0

exp (iξ(1− 2x)) dx =exp(iξ)(1− exp(−2iξ))

2iξ=

sin ξ

ξ

and therefore

sin ξ

ξ=

∫ 1

0

dx

∞∏k=1

exp(iξ2−jrj(x)

)(A.17)

Problem A.1. Use the identity sinα = 2 sin(α/2) cos(α/2) successively to obtain

sin ξ = 2n sin(ξ/2n) cos(ξ/2n) cos(ξ/2n−1) · · · cos(ξ/2)

and then use the above to obtain Vieta’s formula (A.1).

Problem A.2. Prove that∫ 1

0

exp[iξ2−krk(x)]dx = cos(x/2k) . (A.18)

Equations (A.1), (A.17) and (A.18) suggest the validity of

Proposition A.1.∫ 1

0

∞∏k=1

exp[iξckrk(x)]dx =

∞∏k=1

∫ 1

0

exp[iξckrk(x)]dx . (A.19)

Proposition A.1 with ck = 2−k is Vieta’s formula (A.1) written in terms of the

orthogonal (see (A.16)) Rademacher functions.



Probabilistic interpretation We shall provide two proofs of proposition A.1.

Initially, let us remark that the linear combination Ψ(x) =∑nk=1 ckrk(x) is a con-

stant function in each interval In,j = [j/2n, (j+1)/2n), for all j = 0, 1, . . . , 2n−1, of

length |In,j | = 2−n. Observe that the intervals in which rk : k < n assume constant

values ±1 are compatible with the intervals In,j and contain them. Using (A.2) and

(A.10), we may establish a 1-1 relation between each value of j which labels the

interval In,j and the vector ~σ = (σ1, . . . , σn) of components σm ∈ −1, 1:

j/2n =1− σ1

2

1

2+

1− σ2

2

1

22+ · · ·+ 1− σn

2

1

2n. (A.20)

Problem A.3. Derive the dependence of j in ~σ = ~σ(j) given by (A.20) explicitly.

If x ∈ In,j , Ψ(x) is a constant equal to

Ψ(x) = Ψ(j) =

n∑k=1

ckσk = ~c · ~σ

and therefore, for F : R→ R,∫ 1

0

F (Ψ(x))dx =

2n−1∑j=0

|In,j |F (Ψ(j)) =

=1

2n

∑~σ∈(−1,1)n

F (~c · ~σ)

where in the second line we used the relation j ⇔ σ shown in problem A.3 and

(A.20). Substituting F (Ψ) = exp(iξΨ), we obtain∫ 1

0

exp

(iξ

n∑k=1

ckrk(x)

)dx =

1

2n

∑~σ∈(−1,1)n

exp(iξ~c · ~σ)

=

n∏k=1

1

2

∑σ∈−1,1

exp(iξckσ)

=

n∏k=1

cos(ξck) =

n∏k=1

∫ 1

0

exp (iξckrk(x)) dx(A.21)

by (A.18). We conclude the proof of proposition A.1 upon setting ck = 2−k and

then taking the limit n→∞.

q.e.d.

We now introduce the probability space (Ω,B, µ) with Ω = [0, 1), B the Borel

algebra generated by the subintervals of [0, 1), and µ Lebesgue measure on [0, 1).

A random variable (r.v.) f : Ω→ R is a measurable function, i.e., s.t. f−1(A) ∈ Bfor any Borel set A. We see that r1(x), . . . , rn(x) are r.v. defined on the space

([0, 1),B, µ). The probability that a r.v. f assumes some value in the interval

I = [a, b) ⊂ R is the Lebesgue measure of the set

f−1(I) = x ∈ [0, 1) : a ≤ f(x) < b= P(a ≤ f < b)

= µ(x ∈ [0, 1) : a ≤ f(x) < b) = |f−1(I)|



Since the Rademacher functions rk(x) assume only the values ±1,

P(rk = ±1) = µ(x ∈ [0, 1) : rk(x) = ±1) =

∫ 1

0

1∓ rk(x)

2dx = 1/2

Vieta’s formula (A.1) is satisfied due to the following property of the Rademacher

functions. Consider the event Ek that rk(x) assumes the value σk ∈ −1, 1 for

each k = 1, · · · , n individually:

Ek = x ∈ [0, 1) : rk(x) = σk

and the event E(n) that rk(x) assumes the value σk for k = 1, · · · , n jointly:

E(n) = x ∈ [0, 1) : rk(x) = σk for all k = 1, . . . , n =

n⋂k=1

Ek

It follows from the definition of the rk that the probability of these events fulfills

the relation

P(E(n)) = µ(x ∈ [0, 1) : rk(x) = σk, k = 1, · · · , n

=

n∏k=1

µ(x ∈ [0, 1) : rk(x) = σk) =

n∏k=1

P(Ek) =1

2n(A.22)

As an example, in the case n = 3, we may verify by inspection that the Lebesgue

measure of the set of x s.t. r1(x) = 1, r2(x) = −1, r3(x) = −1 is the length of the

interval I3,3 = [3/8, 1/2) (see (??) et ff.):

1/8 = |I3,3| = |I1,0| × |I2,1 ∪ I2,3| × |I3,1 ∪ I3,3 ∪ I3,5 ∪ I3,7| = 1/2× 1/2× 1/2

In chapter ?? subsection ?? we have actually shown (A.22) in general.

(A.22) shows that r1, · · · , rn are (by definition, see, e.g., [6]) independent

r.v., and therefore the integrand on the r.h.s. of (A.21) may be written as the

expectation E (in the language of probability, see again [6]):

E exp

(iξ

n∑k=1

ckrk

)=

∫ 1

0

exp

(iξ

n∑k=1

ckrk(x)

)dx

=∑

~σ∈(−1,1)nexp

(iξ

n∑k=1

ckrk

)µ(x : rk(x) = σk, k = 1, . . . , n)

=∑

~σ∈(−1,1)nexp(iξ~c · ~σ)

n∏k=1

µ(x : rk(x) = σk)

=

n∏k=1

E exp(iξckrk)

which equals the r.h.s. of (A.21), concluding the second proof of proposition A.1.

q.e.d.

By (??), definition A.1 and (A.22) provide an alternative wording (in terms of

Rademacher functions) of (??).



Proposition A.1 expresses a connection between probability theory and elemen-

tary number theory. We now come to the interplay between ergodic theory and

dynamical systems; some connection between ergodic theory and number theory

appears in theorem A.5, and the final paragraphs of the appendix are reserved to

some further elementary notions of number theory which were used in chapter ??.

Given a probability space (Ω,B, µ), and a group Φt; t ∈ R of transformations

with Φt+s = ΦtΦs,Φ0 = I - a flow (resp. semiflow), which preserves the measure

µ:

µ(Φt(A)) = µ(A) for all t ∈ R and A ∈ B

(resp. for all t ∈ R+), we say that A ⊂ B is invariant w.r.t Φt iff Φt(A) = A

for all t ∈ R (resp. t ∈ R+). In case of discrete times, we define T = Φ1, Tn =

Φn1 = Φ1 · · · Φ1 ≡ Φn where the indicates composition, and the · · · refer

to composition n times. In this case, A is invariant if T (A) = A = T−1(A) if T

is an automorphism, i.e., a bijection T : B → B such that µ(A) = µ(T (A)) =

µ(T−1(A)). If T is an endomorphism, i.e., a surjection s.t. for all A ∈ B,

T−1(A) ∈ B and µ(A) = µ(T−1(A)), A is said to be invariant if T−1(A) = A.

Definition A.2. A dynamical system is a group or semigroup of transformations

which preserves the measure µ of a probability space (Ω,B, µ). It is said to be

ergodic if for all A invariant, either µ(A) = 0 or µ(A) = 1.

Sometimes ergodic dynamical systems are also referred to as quasiergodic.

This terminology originates from the following basic theorem:

Theorem A.1 (Birkhoff’s ergodic theorem). If f ∈ L1(Ω, dµ), the “time av-

erage”

limT→∞

1

T

∫ T

0

dtf(Φt(x)) ≡ f(x) (A.23)

exists for almost every x with respect to µ. Moreover, if the system is ergodic, f is

a.e. a constant and thus equal to its “space average”

f(x) =

∫Ω

fdµ (A.24)

a.e. with respect to µ.

For a relatively simple proof, see [54], p. 33. If the system is ergodic, each

integral of motion (i.e., a function g s.t. g (Φt(x)) = g(x)) is constant a.e.. Indeed,

let Ca = x : g(x) < a, then Ca is invariant for all a ∈ R. Hence g(x) < a a.e. or

g(x) ≥ a a.e. Since this is true for all a, g(x) = const. a.e.. This explains the last

assertion of Birkhoff’ theorem. In the discrete case, the time averages are

limN→∞

1

N

N−1∑i=0

f(T ix) (A.25)



Putting f = χA, the characteristic function of A ⊂ B, the time average, i.e., the

mean time spent by the system in region A, equals µ(A).

It may also happen that (A.25) exists for x in some manifold M , and f : M → Ra continuous function, and is independent of the point x in some set B ⊂ M : call

it Ex(f). Then f → Ex(f) defines for all x ∈ B a nonnegative linear operator

on the space C0(M,R) of real continuous functions from M to R, which, by the

Riesz theorem ?? may be written in the form Ex(f) =∫fdµ for a Borel measure

µ on M , for all f ∈ C0(M,R), and for all B ⊂ M , which we assume is of positive

Lebesgue measure (on M). B is the called the ergodic basin of µ, and µ a SRB

measure (Sinai–Ruelle–Bowen measures, see [49], [5]). Note that such a measure

may be “physically observed” by computing time–averages of continuous functions

for randomly chosen points x ∈ M (positive probability of getting x ∈ B). SRB

measures are expected to exist in great generality and are of foremost importance

in nonequilibrium statistical mechanics (see [12] Chapter 13 for a concise but lucid

exposition and further references). In the following exposition, the measure µ may

be thought to stand for a SRB measure.

Definition A.3. An automorphism (or flow) is mixing iff, for all f, g ∈ L2(Ω, dµ),

limn→±∞

∫Ω

f(Tnx)g(x)dµ =

∫Ω

fdµ

∫Ω

gdµ (A.26)

in the case of an automorphism, or

limt→±∞

∫Ω

f(Φt(x))g(x)dµ =

∫Ω

fdµ

∫Ω

gdµ (A.27)

in the case of a flow. Setting f = χA and g = χB , A, B both in B, we obtain

limn→±∞

µ(Tn(A) ∩B) = µ(A)µ(B) (A.28)

which is sometimes used as definition of mixing. In the case of an endomorphism,

the definition is

limn→±∞

µ(T−n(A) ∩B) = µ(A)µ(B) (A.29)

Conversely, one may pass from (A.28) to (A.26) using the fact that finite linear

combinations of characteristic functions are dense in L2(Ω, dµ).

The best “physical” interpretation of the mixing property, e.g., (A.28), has been

given in the famous example of Arnold and Avez [2], a glass containing initially

20 per cent of rum and 80 per cent of coca-cola. Initially, the rum is in region A,

which is disjoint of a region B, fully occupied by coca-cola. After n “mixings” the

percentage of rum in B will be µ(Tn(A) ∩B)/µ(B). In this situation, one expects

that for n→∞ all parts of the glass will contain approximately 20 per cent of the

rum, which is the content of (A.28).

It is very useful to express the mixing property in other, occasionaly more man-

ageable forms. For this purpose, we define:



Definition A.4. The Koopman operator Ut is defined by

(Utf)(x) = f(Φt(x)) (A.30)

for flows, and similarly in the other cases.

Clearly, for each t, Ut is isometric, i.e., preserves the norm in L2(Ω, dµ):

||Utf ||2 =

∫Ω

|f(Φt(x))|2dµ(x) =

∫Ω

|f(Φt(x))|2dµ(Φt(x)) =

∫Ω

|f(x)|2dµ(x) = ||f ||2

and it is a bijection: each g ∈ L2 may be written as some Uf : f(x) = g(Φ−1t (x)).

Therefore, the Koopman operator is unitary. (Φ, µ) is ergodic iff 1 is a simple

eigenvector of U , because f is invariant under Φt if Utf = f : Φ is ergodic iff the

invariant functions are constant a.e.. Since the latter are scalar multiples of one

another, Φ is ergodic iff the subspace of solutions of Uf = f has dimension one.

Related to this, and basic to ergodic theory, is the following mean ergodic theorem

of von Neumann:

Theorem A.2. Let U be a unitary operator on a Hilbert space H, and P be the

orthogonal projection onto Ψ : Ψ ∈ H and UΨ = Ψ. Then, for any f ∈ H,

limn→∞

1

n

n−1∑i=0

U if = Pf . (A.31)

Problem A.4. Prove theorem A.2, using the decomposition

H = Ran(I− U)⊕Ker(I− U†)Ker(I− U†) = Ker(I− U)

Alternatively, see [42], Theorem II.11.

Together with our remarks preceding theorem A.2, we have shown that T is

ergodic iff, ∀f, g ∈ L2(Ω, dµ),

limn→∞

1

n

n−1∑i=0

(U if, g) = (f, 1)(1, g) (A.32)

The assertion in the form corresponding to (A.26) is

limn→∞

1

n

n−1∑i=0

∫f(T ix)g(x)dµ =

∫fdµ

∫gdµ (A.33)

or, in the continuum case,

limT→∞

1

T

∫ T

0

∫f(φt(x))g(x)dµ(x) =

∫fdµ

∫gdµ (A.34)

We now come to the important problem of relating the spectrum of U on 1⊥to ergodic properties. We need a final definition:



Definition A.5. We say that an endomorphism T is weak-mixing iff

limn→∞

1

n

n−1∑i=0

|µ(T−i(A) ∩B)− µ(A)µ(B)| = 0

with analogous definitions in the other cases (automorphism, flow). As usual, equiv-

alent definitions similarly to (A.26) and (A.27) follow, and therefore T is weakly

mixing if, ∀f, g ∈ L2(Ω, dµ),

limn→∞

1

n

n−1∑i=0

|(U if, g)− (f, 1)(1, g)| = 0

We now have:

Theorem A.3. If T is an automorphism and U the corresponding Koopman oper-

ator, T is weak mixing iff 1 is the only eigenvalue of U , and on 1⊥ the spectrum

of U is continuous.

Problem A.5. Prove theorem A.3 using the spectral theorem for U and Fubini’s

theorem, in a very similar fashion to the proof of Wiener’s theorem ??.

For a mixing automorphism, (A.26) may be written in terms of the Koopman

operator U as

limt→±∞

(Utf, g) = (f, 1)(1, g) for all f, g ∈ L2(Ω, dµ) (A.35)

As a consequence of (A.35), the Riemann-Lebesgue lemma ?? and the spectral

theorem we have (see [42], Theorem VII.15, p. 241):

Theorem A.4. Let T be an automorphism and U the associated Koopman operator.

If U has purely a.c. spectrum on 1⊥, then T is mixing.

We have mentioned previously the “physical” significance of the mixing property.

We now inquire into its relation with the ergodic property and the related signif-

icance in a real physical framework, that of statistical mechanics. It is clear from

(A.28) or (A.29) that mixing implies ergodicity, for, take A invariant, T−1(A) = A,

and let B = A in (A.29), for instance, then the latter yields µ(A) = µ(A)2 and

therefore µ(A) = 0 or µ(A) = 1, i.e., the system is ergodic. The converse does not

hold, however, as we now show. Let M be the circle z ∈ C : |z| = 1, µ Lebesgue

measure over M , Φ the translation Φ(z) = θz with θ = exp(2πiω) where ω ∈ R.

Consider the orthonormal basis of L2(M,dµ) given by zp : p ∈ Z; we have

(Unzp, zp) = θpn (A.36)

Φ is ergodic iff 1 is a simple eigenvalue of U , i.e., iff pω is not in Z if p 6= 0, i.e.,

if ω is irrational. However, Φ is not mixing under the latter condition, because,

taking f = g = zp in (A.35), we obtain, for p 6= 0, limn→±∞(Unzp, zp) = 0 as the

condition for mixing, but, by (A.36), it is not satisfied because limn→±∞ θpn does



not exist. These results extend immediately to higher dimensional tori and show

that ergodicity does not imply “approach to equilibrium”: the translations of the

torus do not deform a region A, they are such that the intersections of Tn(A) with

B are alternately empty or of positive measure. We now show that mixing does

imply approach to equilibrium in the framework of classical statistical mechanics.

Consider the dynamical system generated by the motion of N matter points (gas

molecules) in a fixed volume V , and let Γ = x ≡ (qi, pi), i = 1, . . . , 3N be the

corresponding phase space, with qi denoting the generalized coordinates, and pi the

momenta. If the system is isolated (fixed total energy E), Γ will be compact and

there exists [51] an invariant measure µ which may be interpreted as the distribution

in thermodynamic equilibrium. Starting from initial conditions in a certain volume

V1, the set of admissible coordinates and momenta of the molecules forms a subset

A ⊂ Γ. The formula

µ0(B) =µ(A ∩B)

µ(A)(A.37)

defines a initial distribution (which is not the equilibrium distribution), interpreted

as conditional distribution relatively to the system’s known initial condition. We

may now relate µ0 to the knowledge of the state of the system at time t by the

measure µt defined by

µt(B) = µ0(Tt(B)) (A.38)

where T is the flow which leaves µ invariant. In the given example,

µ(F ) = µE(F ) =

∫F

δ(H(x)− E)dx (A.39)

the microcanonical Gibbs measure where H(x) is the Hamiltonian describing

the system and (A.38) is Liouville’s theorem. Suppose, now, that µ0 is a.c. with

respect to µ, i.e., there exists ρ0 ∈ L1(M = Γ, dµ) such that

dµ0 = ρ0(x)dµ . (A.40)

Then,

µt(B) =

∫M

χB(Ttx)dµ0 =

∫M

χB(Ttx)ρ0(x)dµ (A.41)

If the system is mixing, we obtain from (A.35), as t→∞,

limt→∞

µt(B) =

∫M

χBdµ

∫M

ρ0dµ = µ(B)1 = µ(B) (A.42)

which means that, whatever the initial distribution µ0, normalized and a.c. w.r.t.

µ, the time–translates µt of µ0 under Tt converge, for t→∞, to the equilibrium dis-

tribution. (A.42) may be taken as the definition of the approach to equilibrium .

It means that mixing systems are “memoryless”, i.e., they posess a stochastic char-

acter which justifies equilibrium statistical mechanics. The microscopic mechanism

of this process of loss of memory is the sensitive (exponential) dependence on initial



conditions (later on precisely defined) produced by “defocalizing shocks” between

the gas molecules, first pointed out by Krylov, see [50] and references given there.

It may also be felt in the exponential rate at which the mixing condition takes

place, a topic to which we also return later.

We may picture the gas molecules as a system of hard spheres enclosed in a cube

with perfectly reflecting walls or periodic b.c.. This is supposed to be a K–system

(see [54], definition 4.7, p. 101). Rising still one step in this so–called ergodic

hierarchy (see [34]), we come to Bernoulli systems , such as the one we presently

introduce along the lines of [2].

Define Tr by Tr : X → Y , where X = [0, 1] and Y = [0, 1], by

Trx = fr(rx) ≡ rx mod1 (A.43)

Note that this mapping is not 1-1, it is an endomorphism. In fact, if r = 2, we

see that the inverse image of a point x is x/2 or (x + 1)/2. Tr is called the r–adic

transformation and leaves Lebesgue measure µ invariant. Indeed, if

A = [m

rn,m+ 1

rn] for m ∈ [0, rn − 1] we have

T−1r (A) =

r−1⋃s=0

[m+ s

rn+1,m+ s+ 1

rn+1

]and

µ(A) = r−n = µ(T−1r (A)) = rr−(n+1)

Consider the case r = 2. T2 has a simple alternative description: writing x =

.ε1ε2 · · · , i.e., x =

∞∑n=1

εn2n

with εn ∈ 0, 1 as in (A.5), then T2x = .ε2ε3 · · · , Tn2 x =

.εn+1εn+2 · · · (check!) and therefore T2 is also known as the one-sided shift. Let

M = [0, 1]Z be the probability space. The sigma–algebra of subsets of M generated

by the cylinders C = (xn)n∈Z : xi = εis , s = 1, . . . , k, where is is an increasing

finite sequence of integers, and εis ∈ 0, 1 coincides with the Borel sigma–algebra

of M , equipped with the product topology. If µ0 is a probability measure s.t.

µ0(0) = p1 and µ0(1) = p2, with p1 + p2 = 1, p1 > 0, p2 > 0, we take as µ the

product measure; in particular, µ(C) =∏ks=1 pis . The mixing property (A.29) need

only be verified for A, B cylinders. But, for n sufficiently large, T−n2 (A) and B are

disjoint and, since µ(T−n2 (A)) = µ(A), we have that µ(T−n2 (A) ∩ B) = µ(A)µ(B)

for n sufficiently large. Thus we have proven that T2 is mixing.

Take, now, as our dynamical system the set of autonomous n first order difference

equations

~xn+1 = T (~xn) (A.44)

where ~x ≡ (x1, . . . , xn), T (~x) ≡ (T1(~x), . . . , Tn(~x)). The mapping T : M → Rn,

where M is a manifold (which we often identify with an open set in Rn) is a

vector field: we assume it is an automorphism. Let J(~x) denote the Jacobian



matrix Ji,j(~x) =∂Ti∂xj

(~x) associated with (A.44). Defining Tn(~x) ≡ T (T · · ·T ) (~x)

as before and using the chain rule we have

J(Tn)(~x) = J(Tn−1)(~x) · · · J(T )(~x)J(~x) (A.45)

Let µ denote the measure invariant under the automorphism T , which completes

the definition of the dynamical system. We shall assume it has compact support

and that it is ergodic under the automorphism T . Then, Oseledec’s multiplicative

ergodic theorem ([39], see also [15]) implies the existence, for almost all ~x with

respect to µ, of the limit

limn→∞

[J(Tn)†(~x)J(Tn)(~x)] ≡ Λ~x (A.46)

The positive matrix Λ~x has eigenvalues λ1 > λ2 > · · · .

Definition A.6. λi, with multiplicity mi, is called a Lyapunov exponent.

By ergodicity of µ it follows that these exponents are constants for almost all

values of ~x (for SRB measures this will hold for almost all ~x in the ergodic basin

of µ). If Ei~x is the subspace of Rn corresponding to the eigenvalues smaller than

exp(λi), Rn = E1~x ⊃ E2

~x ⊃ · · · , one can show that for almost all ~x with respect to

µ,

limn→∞

log ||J(Tn(~x))u||n

= λi for i = 1, 2, . . . (A.47)

if u ∈ Ei~x\Ei+1~x . In (A.47) the norm is the Euclidean norm in Rn. In particular, for

all u ∈ Rn\E2~x, the limit equals the largest Lyapunov exponent λ1. We shall see

explicit examples shortly. But note that existence of a positive largest Lyapunov

exponent is a precise definition of what is meant by sensitive or exponential depen-

dence on initial conditions, which is an important element of chaotic behavior: see

[15] for a detailed discussion. It is however easy to illustrate the phenomenon on

the basis of the example T2: consider two x1, x2 both in [0, 1], close in the sense that

the first n digits are identical. By the interpretation of T2 as one-side shift, it fol-

lows immediately that Tn2 x1 and Tn2 x2 differ already in the first digit: an initially

exponentially small difference 2−n is magnified by the evolution to one of order

O(1). This is the best way to approach exponential sensitivity to initial conditions

for quantum systems: see chapter ??.

It turns out, however, that T2 has more than sensitive dependence on initial

conditions: the sequence of iterations Tn2 x0 has, for almost every x0, the same

random character as the sequence of successive tossings of a coin; this is what

characterizes Bernoulli systems and places them at the top of the ergodic hierarchy:

Theorem A.5 (Borel’s theorem on normal numbers). For a.e. x0 ∈ [0, 1],

the n–th digit in its binary expansion has relative frequency 1/2, i.e., almost all x0

is normal to base 2.



Proof. Let f : [0, 1]→ [0, 1] be defined by

f(x) =

0 if x < 1/2 ,

1 if x ≥ 1/2 ,

Since T2 is ergodic (being mixing),

limn→∞

1

n

n−1∑i=0

f(T i2x) =

∫ 1

0

f(x)dx = 1/2

by Birkhoff’s theorem A.1, for a.e x ∈ [0, 1], but

f(T i2x) =

0 if εi = 0 ,

1 if εi = 1 ,

which proves the theorem.

q.e.d.

Problem A.6. Which numbers comprise the set of zero Lebesgue measure in theo-

rem A.5?

The two-sided shift (i.e. in both directions) is a.e. an automorphism of [0, 1]→[0, 1]: the baker’s map [32]. It is a caricature of the Smale horseshoe, which describes

the local behavior of Hamiltonian systems near homoclinic points (which we shall

define shortly): see [2, 9]. We now introduce it because it plays an important

illustrative role in chapter ??.

Let X be the unit square X = [0, 1] × [0, 1]. The Borel sigma algebra B is

now generated by all possible rectangles of the form [0, a] × [0, b], and the Borel

measure µ is the unique measure on B such that µ([0, a]× [0, b]) = ab. We define a

transformation S : X → X - the baker’s transformation by

S(x, y) =

(2x, y/2) if 0 ≤ x < 1/2 and 0 ≤ y ≤ 1 ,

(2x− 1, y/2 + 1/2) if 1/2 ≤ x ≤ 1 and 0 ≤ y ≤ 1 ,(A.48)

The reader might wish to look at fig. 5 of [34] or verify for himself the effect of S

on some figure symmetrically disposed w.r.t. the line x = 1/2: as in the kneading

of a piece of dough, one first squashes the unit square to half its original height and

twice its original width, and then cuts the resulting in half and moves the right half

of the rectangle above the left. Repeating this operation several times, the result is

to very quickly scramble, or mix, various parts of the original figure: this provides

a very colourful illustration of the mixing property!

The Koopman operator, defined by (A.30), is unitary from L2(Ω, dµ) to itself,

and leads (for automorphisms) to a characterization of the mixing property by

(A.35). It is interesting, from the point of view of approach to equilibrium, to con-

sider an alternative description. Consider, to fix ideas, the case of endomorphisms,

and define the Koopman operator (we shall keep the same name for simplicity) now

as an operator from L∞(Ω, dµ) to L∞(Ω, dµ) by (see [32], p. 42):

(Uf)(x) = f(Tx) (A.49)



It is well–defined because f1(x) = f2(x) a.e. implies f1(Tx) = f2(Tx) a.e. by the

definition of endomorphism. It has the property

||Uf ||∞ ≤ ||f ||∞ (A.50)

If f ∈ L1, the functional

g ∈ L∞ → (f, Ug) (A.51)

defines a continuous linear functional on L∞, with

|(f, Ug)| ≤ ||f ||1||g||∞ (A.52)

We may thus define a bounded linear operator - the Ruelle-Perron-Frobenius

operator (see e.g. [45, 32]) P : L1 → L1 by

(Pf, g) = (f, Ug) (A.53)

From (A.53), for g = χA, where A is any Borel subset of Ω, we obtain∫A

(Pf)(x)dµ(x) =

∫T−1(A)

f(x)dµ(x) (A.54)

Conversely, (A.54) defines P uniquely by the Radon-Nikodym theorem, see [32],

p.37. In [32], P is mentioned as being the adjoint of the Koopman operator U

(defined by (A.49)), but the Banach space adjoint (see, e.g., [42], p. 185) is an

operator defined from the dual space (the space of continuous linear functionals on

the given space) to the dual space: in the present case, the dual of L∞ is a huge

space, which contains L1 properly. In fact, by exercise 8 (b) of [42], p. 86, there

exists a bounded linear functional λ on L∞(R) such that λ(f) = f(0) ∀f ∈ C(R),

i.e., the dual space of L∞(R) contains the Dirac measure at the origin. But, as

we shall see, it is absolutely crucial for the developments here and in chapter ??

that P be considered as an operator mapping densities (in contrast to individual

orbits, formally characterized by the evolution of delta measures at some point) to

densities!

The baker transformation is a.e. invertible (it is not invertible on the line y =

1/2): it is only a.e. a diffeomorphism. Taking A = [0, x] × [0, y] we have for

0 ≤ x ≤ 1 and 0 ≤ y < 1/2, T−1(A) = [0, x/2]× [0, 2y] and thus, by (A.54),

(Pf)(x, y) =∂2

∂x∂y

∫ x/2

0

ds

∫ 2y

0

dtf(s, t) = f(x/2, 2y) , if 0 ≤ y < 1/2 (A.55)

and for 1/2 ≤ y ≤ 1 and 0 ≤ x < 1

T−1(A) = [0, x/2]× [0, 1] ∪ [1/2, 1/2 + x/2]× [0, 2y − 1]

and hence, again by (A.54),

(Pf)(x, y) =∂2

∂x∂y

[∫ x/2

0

ds

∫ 1

0

dtf(s, t) +

∫ 1/2+x/2

1/2

ds

∫ 2y−1

0

dtf(s, t)

]= f(1/2 + x/2, 2y − 1) , if 1/2 ≤ y ≤ 1 (A.56)



Note that, by (A.54), the Ruelle–Perron–Frobenius operator maps densities to

densities: a density (function) f any f : f ∈ L1 and f ≥ 0 a.e.. We denote the

class of density functions by D. The positivity preserving character of P , i.e.,

Pf ≥ 0 a.e. if f ≥ 0 a.e. (A.57)

is immediate from the Radon-Nikodym theorem (see, again, [32], p. 37). It leads

to the inequality

||Pf ||1 ≤ ||f ||1 (A.58)

Let f+(x) ≡ max0, f(x) and f−(x) ≡ min0, f(x). We have that (Pf)+ =

(Pf+−Pf−)+ ≤ (Pf)+ and (Pf)− ≤ Pf−. From these inequalities it follows that

|Pf | ≤ P |f |, and it is now simple to show:

Problem A.7. Prove (A.58).

Analogously to problem A.7, it may also be proved that equality in (A.58), i.e.,

||Pf ||1 = ||f ||1, occurs iff Pf+ and Pf− have disjoint supports; in particular,

||Pf ||1 = ||f ||1 if f ≥ 0 a.e. (A.59)

Properties (A.57) and (A.59) define a Markov operator. If P is a Markov operator

and for some f ∈ L1, Pf = f , then f is called a fixed point of P . If Pf = f , then

Pf+ = f+ and Pf− = f− (see [32], proposition 3.1.3). If f ∈ D, and Pf = f , f is

called a stationary density of P . We have the following important theorem ([32],

theorem 4.4.1):

Theorem A.6. T is mixing iff Pnf is weakly convergent to 1 for all f ∈ D, i.e.,

limn→∞

(Pnf, g) = (f, 1)(1, g) ∀f ∈ D and ∀g ∈ L∞ . (A.60)

If P is a Markov operator with stationary density 1, i.e.,

P1 = 1 (A.61)

then Pnf converges weakly to the stationary density: the weak limit is a fixed

point of P and, being unique, it is the function f = 1. This is an approach to

equilibrium in the formerly defined sense.

How does all this apply to the baker’s transformation? Property (A.57) and

hence (A.59) follow from (A.55,b). We now consider the mixing property. The latter

is simplest to see by noting that, if (x0, y0) ∈ [0, 1] × [0, 1], s.t. x0 = .a1a2a3a4 · · ·and y0 = .b1b2b3b4 · · · , by justaposing the representations in the following way:

· · · b4b3b2b1a1a2a3a4 · · · , the baker transformation may be seen to correspond to

the shift of the decimal point to the right - the so–called two–sided shift, see [54],

p.18 (check!). Then mixing follows by the same proof as for the one–sided shift.

Finally, (A.61) also follows from (A.55,b). Thus, by theorem A.6, there is approach

to equilibrium for the baker’s map. Note that f ∈ L1 excludes a delta measure at

some point in the square: this is expected because the baker map is (a.e.) invertible,



and individual orbits are therefore reversible. We see hereby that approach to

equilibrium in the theory of dynamical systems in the sense defined above depends

on two key features: Q1) One looks only at the evolution of densities, and not

individual orbits: this is a reduced description of the system, similar to Boltzmann’s

approach (see chapter ??); Q2) the initial state (density) is not the (invariant,

stationary) density, which is the uniform distribution on the unit square. We return

to these points in chapter ?? section ??.

We finish this appendix with two subjects: Anosov systems and rate of mixing

(with some remarks on non-uniformly hyperbolic systems) and a few basic results

of number theory used in the main text.

The baker transformation is a prototype of an important class of transforma-

tions, the Anosov systems [1]. Another special example of Anosov system is the

Arnold cat map [2]:

TA(x, y) = (x+ y, x+ 2y) mod 1 (A.62)

The fixed points F of TA are given by (x + y, x + 2y) = (x, y) hence F = (0, 0).

This is an example of a hyperbolic point. In the more general case (A.44), but

now with T a mapping from a complex manifold M to Cn, let ~x0 be an invariant

or equilibrium point of the vector field T :

Definition A.7. A linear vector field on Cn given by ~x→ L~x, where L is a n× nmatrix with complex entries, is said to be hyperbolic iff the spectrum σ(L) of L has

empty intersection with the imaginary axis. An equilibrium point ~x0 of the vector

field T is said to be hyperbolic iff the Jacobian matrix J( ~x0) is a hyperbolic vector

field. By the Jordan decomposition (see [20]), it may be shown that σ(exp(L)) =

exp(σ(L)): hence, ~x0 is hyperbolic iff J( ~x0) does not have any eigenvalue of unit

modulus.

Due to the Hartman–Grobman theorem (see [19]) the phase portrait of a vector

field near a hyperbolic point has the same topological structure as its linearization

at this point: therefore a hyperbolic fixed point of a vector field is unstable. Such

is not the case for elliptic fixed points, for which J( ~x0) has modulus one: their

stability depends on the character of the non-linear terms [19]. The Hartman–

Grobman theorem also guarantees the existence of the so–called local (stable and

unstable) manifolds at any hyperbolic point (see also [50], lecture 17). We now

return to the cat map. The Jacobian matrix(1 1

1 2

)has eigenvectors v± given, respectively, by: 1

1±√

5

2



corresponding to eigenvalues r± =(3±√

5)/2. The corresponding invariant (local)

manifolds are W±(~0)

= E±~0 with E± generated by v±, with angular coefficients

s and −s−1, with s = (1 +√

5)/2. Since the latter are irrational numbers, these

manifolds cover the torus densely. Because r+ > 1, (0, 0) is a hyperbolic point.

Problem A.8. Identify the sets E1~0

and E2~0

in the definition of Lyapunov exponents

and compute them for the cat map.

The sum of the Lyapunov exponents for an area preserving map, such as the

cat map above, is zero: this follows in a straightforward way from their definition.

In Hamiltonian and other measure preserving maps chaos has its origins in homo-

clinic points [2] defined as points of intersection of the local (stable and unstable)

manifolds associated to a given hyperbolic point, other than the point itself (see also

[9]). The usual proofs of the existence of the “homoclinic tangle” rely on the (often

verified, such as in the cat map) hypothesis that the manifolds intersect transver-

sally, i.e., are not tangential at the homoclinic point (Melnikov’s method, see [21]

for a pedagogic discussion). See also the monograph of Palis and Takens for the

discussion of homoclinic tangencies [40]. In the case of the cat map, considering the

image of a region B after a large number n of iterations of the map TA, it will consist

of the product of rn+ expansions in the unstable direction v+, and rn− contractions in

the stable direction v−, i.e., a highly ”stretched” band in direction v+ (mod 1): for

n large TnAB covers the torus and the mixing property holds (see [32] Example 4.4.3

p. 71 for the proof). The two invariant manifolds intersect in homoclinic points;

the image of a homoclinic point is, by definition, also a homoclinic point, and, thus,

the cat map posesses an infinity of homoclinic points; it is also immediate (check!)

that the manifolds intersect transversally at these points.

Problem A.9. Which points of the unit square yield periodic orbits of the cat map?

What can one say about the asymptotic distribution of their periods?

Hint for the second part: see the discussion in [9] on the Ozorio–Hannay uni-

formity principle.

Anosov systems such as the cat map (A.62) are “uniformly hyperbolic”. For

such systems there is exponential decay of correlations∣∣∣∣∫Ω

f(Tnx)g(x)dµ−∫

Ω

fdµ

∫Ω

gdµ

∣∣∣∣ ≤ Crn for some r < 1 (A.63)

(see [53] for a nice review). Much of the recent work in dynamical systems has been,

however, devoted to systems displaying only a weak form of hyperbolicity. One of

them is that T is expanding in its critical orbit, see theorem 5.1, p. 125, of [53]

or the original early article of Baladi and Viana [4] for a readable account for an

important class of models. Together with the other assumptions, it is proved that

T admits a unique absolutely continuous invariant measure which is ergodic (and



so is an SRB measure for T ) and exhibits exponential decay of correlations in the

sense of (A.63).

Physically more “realistic” systems remain, however, very difficult to analyse.

A prototype of the latter is the famous standard or Chirikov-Taylor map ([50],

p. 138), which is the classical analogue of the kicked rotor (??), (??): it is a map

of the cylinder R× T to itself, defined by:

zn+1 = zn + k sin(2πφn)

φn+1 = φn + zn+1 mod1 (A.64)

The importance of this map is reviewed by Sinai in [50], p. 138. There exist very

few rigorous results for this map: the beautiful Aubry–Mather theory shows that

for k > 1 the standard map has no invariant curves which may be represented by

a continuous function z = f(φ) (for a simple proof of this, see [50], p. 142): the

structures occurring for k > 1 are fractal objects denominated Cantori. Another

result, much more difficult, due to Duarte [13, 14] proves, very roughly speaking,

the existence of an abundance of of elliptic islands even deep inside the region k > 1

of “hard chaos” in the model (actually this region is conjctured to be k > kcr with kcr

given by Greene’s conjecture, see the discussion in [50]). Duarte’s remarkable result

throws considerable doubt on the validity of exponential decay of correlations (A.63)

in the chaotic region: it may be algebraic instead, see chapter ?? for further remarks.

Estimates on the rate of decay of correlations are among the most challenging open

problems, both in the theory of dynamical systems and in the theory of “quantum

chaos”.

We conclude this appendix with a brief review of some of the most fundamental

aspects of number theory used in the main text, in particular chapter ?? sections ??

and ??. The reader will find a complete and very readable account of these features

in Khinchin’s book [30].

Irrational numbers may be approximated by rationals with arbitrary precision.

For example, π is approximated by the sequence

s/r = 3/1, 31/10, 314/100, 3142/1000, 31416/10000, · · ·which yields the better approximants the greater the value of r. For such decimal

approximations, we have for an irrational number µ:

∣∣∣∣µ− snrn

∣∣∣∣ < 1/rn, with rn =

10n and a similar inequality is valid for any base. These approximants are not,

however, the best possible, because they are contaminated by the intrinsic properties

of the base. We are thus led to search a sequence of approximants µnn≥1 which

is independent of the base: such a sequence is proportionated by the continued

fractions:

µn = a0 +1

a1 +1

a2 + · · ·

(A.65)

The approximants in (A.65):

µn = sn/rn (A.66)



obey the recursion relations

sn = ansn−1 + sn−2 for n ≥ 2

rn = anrn−1 + rn−2 for n ≥ 2 (A.67)

In order to prove (A.67) by induction, suppose it valid up to order (n−1). Defining

s′n−1

r′n−1

= a1 +1

a2 +1

a3 +1

· · ·+ 1

an

we have:

sn = a0s′n−1 + r′n−1

rn = s′n−1 (A.68)

and by the induction hypothesis

s′n−1 = ans′n−2 + s′n−3

r′n−1 = anr′n−2 + r′n−3 (A.69)

Introducing (A.69) into (A.68)

sn = a0(ans′n−2 + s′n−3) + anr

′n−2 + r′n−3

rn = ans′n−2 + s′n−3

or

sn = an(a0s′n−2 + r′n−2) + a0s

′n−3 + r′n−3 = ans

′n−1 + s′n−2

rn = anr′n−1 + r′n−2

Clearly (A.67) is valid for n = 2 and thus the result is demonstrated.

The difference between two approximants µn and µn−1 is

sn−1

rn−1− snrn

=sn−1rn − snrn−1

rnrn−1=

(−1)n

rnrn−1(A.70)

In fact, multiplying the first formula in (A.67) by rn−1, the second one by sn−1 and

subsequently subtracting the resulting first one from the resulting second one, we

obtain

sn−1rn − rn−1sn = −(sn−2rn−1 − rn−2sn−1)

and, since (s−1 = 1, r−1 = 0), r0s−1 − s0r−1 = 1, and therefore it follows that

sn−1rn − rn−1sn = (−1)n

from which (A.70) follows.

Problem A.10. Show that for all n ≥ 2,

sn−2

rn−2− snrn

=(−1)n−1anrnrn−2



The two results above show that the convergents of even order form an increas-

ing sequence (we assume an > 0 for all n > 1), and those of odd order form a

decreasing sequence. It follows therefore that both sequences have the same limit.

Indeed, since, by (A.67), each convergent of odd order is greater than the immedi-

ately subsequent convergent of even order, it follows that each convergent of odd

order is greater than each convergent of even order, and that the two sequences of

convergents (of odd and even order) tend to the same limit.

The above result culminates (see [30]) in

Theorem A.7 (Hurwitz’s theorem). For any µ, there exist rational approxi-

mants sn/rn such that ∣∣∣∣µ− snrn

∣∣∣∣ ≤ 1√5r2n

. (A.71)

Clearly the approximants converge the quicker the an diverge (for a rational

number, some an = ∞ after a n = n0 finite). As an illustration, we suggest that

the reader show the explicit results for π (precise up to the kn–th digit):

s0/r0 = 3

s1/r1 = 22/7 ' 3.1429 (k1 = 2)

s2/r2 = 333/106 ' 3.1415 (k2 = 3)

s3/r3 = 355/113 ' 3.1515929 (k3 = 6)

(the last result, according to M.Berry, was known to Lao–Tse (604–531 b.C.)). The

slowest convergence, corresponding to the irrational number most poorly approxi-

mated by rationals, corresponds to the golden mean: a0 = a1 = · · · = 1,

1 +11

1 + · · ·

=

√5− 1

2.

This number is important for several reasons: it appears in problem ?? in con-

nection with the P.V. numbers, and is also conjectured to be the rotation number

corresponding to the critical coupling kcr in the standard map, which signals the

disappearance of the last “KAM torus”, i.e., in the standard map the last continuous

invariant curve. It saturates the inequality (A.71) provided by Hurwitz’s theorem.

This last remark brings us to our last topic: diophantine numbers.

Definition A.8. An irrational number µ is said to be diophantine of type σ iff

there exists a γ > 0 such that∣∣∣µ− s

r

∣∣∣ ≥ γ

rσfor all s/r ∈ Q (A.72)

In passing, we note that µ is said to be a Liouville number if it is neither rational

nor diophantine. Alternatively, µ is Liouville iff there exists a sequence sn/rn ∈ Q

such that

∣∣∣∣µ− snrn

∣∣∣∣ < r−nn for all n ≥ 1. By a theorem due to Liouville, every



algebraic number (zero of a polynomial with rational coefficients) is diophantine.

On the other hand, the number µ =

∞∑n=1

2−n! is by definition Liouville. Thus there

exist non-algebraic numbers: they are the transcendental numbers. For much more

about this, see [38].

The set of diophantine numbers of type σ has full Lebesgue measure if σ > 2:

for almost all µ ∈ [0, 1] and for all τ > 0, there exists a number K = K(µ, σ = τ+2)

such that ∣∣∣µ− s

r

∣∣∣ ≥ K

r2+τ(A.73)

for all s/r ∈ Q. In order to show (A.73), fix r and s. Consider the set of numbers

µ ∈ [0, 1] such that (A.73) is violated. They form a set of length ≤ 2K

r2+τ, and,

since s ≤ r, the union of all such intervals with the same r has measure ≤ 2K

r1+τ.

Summing over all the r we obtain a set with Lebesgue measure ≤ CK, where

C = 2∑∞r=1 r

−1−τ <∞ since τ > 0. Choosing K arbitrarily small, we may render

the measure of the set of numbers violating (A.73) as small as wished. Thus, the

set of numbers violating (A.73) for all K has zero Lebesgue measure.


Appendix B

Transfer matrix, Prufer variables andspectral analysis of sparse models

Weyl–Titchmarsh m–function of a Jacobi matrix We may think of a Ja-

cobi matrix

J =

v1 p1 0 0 · · ·p1 v2 p2 0 · · ·0 p2 v3 p3 · · ·0 0 p3 v4 · · ·...

......

.... . .

, (B.1a)

as an operator acting on the space of complex–valued square–summable sequences

u = (un)n≥0, denoted by l2(Z+), and the Schrodinger equation associated with J

reads

((J − zI)u)n = pnun+1 + pn−1un−1 + (vn − z)un = 0 (B.1b)

for n ≥ 1 with p0 ≡ 1 and z ∈ C. A Jacobi matrix Jφ is said to satisfy a φ–boundary

at 0 if

u0 cosφ− u1 sinφ = 0 , (B.1c)

for some φ ∈ [0, π); (B.1a), as an operator, satisfies Dirichlet 0–boundary phase

condition u0 = 0 and we write J = J0. The phase boundary plays an important

role on the characterization of singular part of the spectrum, as we shall see in the

following. We call J an admissible Jacobi matrix if it is of the form (B.1a) and

satisfies 0 < pn ≤ 1 with∑n≥0 p

−1n = ∞ and |vn| ≤ M < ∞. Sooner, we shall

restrict the admissible class to two sparse models dealt in this monography (see ??)

et seq.): (i) pn = 1 ∀n and vn = 0 except for a lacunary subsequence A = (aj)j≥1;

(ii) vn = 0 ∀n and pn = 1 except for a lacunary subsequence A = (aj)j≥1.

The set of all solutions uφ(z) of (B.1b) and (B.1c) form a two–dimensional vector

space whose base: y(z) = uα(z) and w(z) = uα+π/2(z) may be chosen by fixing the

initial conditions

y0 = sinα , y1 = cosα

w0 = cosα , w1 = − sinα (B.2)

23



for some α ∈ [0, π). For =z 6= 0, there are two alternatives: either we have the

limit–circle (in which case both y(z), w(z) ∈ l2(Z+)) or the limit–point case, for

which there exist exactly one linear independent l2(Z+)–solution. We have

Proposition B.2. If J is an admissible Jacobi matrix, then J is limit–point.

Proof. Let y(z) and w(z) be a base for (B.1b). By Green’s formula (see e.g. (1.20)

of [52])

0 = (w, Jy)n − (Jw, y)n = −Wn[y, w] +W0[y, w] (B.3a)

where Wn[f, g] is the (weighted) Wronskian function:

Wn [f, g] = pn(fngn+1 − fn+1gn) (B.3b)

of two sequences f = (fn)n≥0 and g = (fn)n≥0 at n and (f, g)N =

N∑n=1

fngn is the

inner product restricted to 1, . . . , N. (B.3a) implies that the Wronskian of two

linearly independent (L.I.) solutions of the Schrodinger equation is constant and

does not vanish:

Wn[y, w] = W0[y, w] = p0 (y0w1 − y1w0) = −1 (B.3c)

by (B.2). Together with (B.3b) and the Schwartz inequality, we have

∞∑n=0

1

pn=

∞∑n=0

−1

pnWn[y, w] =

∣∣∣∣∣∞∑n=1

(ynwn+1 − yn+1wn)

∣∣∣∣∣ ≤ 2 ‖y‖ ‖w‖ (B.3d)

where ‖·‖ is the l2(Z+)–norm. If J is admissible, then the l.h.s. of (B.3d) diverges

and necessarily one of the two L. I. solutions: y or w does not belong to l2(Z+),

concluding the proof.

q.e.d

We set α = 0 in the following propositions and refer to chapter 2 and appendix

B of [52] for references and further directions.

Proposition B.3. The l2(Z+)–solution of (B.1b) with =z > 0 can be written as

χ(z) = w(z)−m(z)y(z) (B.4)

where m(z) = m0(z), the Weyl-Titchmarsh m–function related to Jα with α =

0, is defined by

m(z) :=

(e1,

1

J0 − zIe1

)(B.5)

and ej , j ≥ 0 denotes the canonical base of l2(Z+): (ej)i = δij.

Proof. Let χ(z) be the l2(Z+)–solution of (B.1b). Since y(z) is not an eigenvector

of J when =z > 0, it cannot be an l2(Z+)–solution of (B.1b) so, χ(z) is a linear


Transfer matrix, Prufer variables and spectral analysis of sparse models 25

combination of y(z) and w(z) of the form (B.4). The Green’s function associated

with J

gij(z) =

(ei,

1

J − zIej

)can be written in terms of the two L.I. solutions of (B.1b) y(z) and χ(z):

gij(z) =

yi(z)χj(z)

−Wj [y, χ]if i < j

yj(z)χi(z)

−Wj [y, χ]if j < i

. (B.6)

Setting i = j = 1 in (B.6), together with Wj [y, χ] = W0 [y, w] = −1 and

y1(z)χ1(z) = m0, the proof is concluded.

q.e.d.

Remark B.1. Since Jφ is a rank–one perturbation of J0:(Jφu

)n

=(J0u

)n

+

δ0,nu0 tanφ, the Weyl–Titchmarsh m–function mα(z), defined by (B.5) with J0

replaced by Jα, satisfies

mα(z) =m(z)

1−m(z) tanα. (B.7)

Definition (B.5) implies that m(z) is holomorphic in C\σ(J) and, since it maps

the upper–half plane H into itself, it is, in addition, a Herglotz function (called

also Pick or Nevanlinna–Pick function).

Problem B.11. Prove the above statements.

Hint. Use the first resolvent equation (J − zI)−1 − (J − ζI)−1= (z −

ζ) (J − zI)−1(J − ζI)−1

for the former.

As a consequence, m(z) admits a unique canonical integral representation (the-

orem I, chapter 2 of [24]):

m(z) = az + b+

∫ ∞−∞

(1

λ− z− λ

λ2 + 1

)dρ(λ) , =z > 0

where a ≥ 0, b ∈ R are constants and dρ(λ) is a Borel–Stieltjes measure such that∫ ∞−∞

dρ(λ)

λ2 + 1<∞ .

It follows from (B.5) that

m(z) = −z−1 +O(z−2)

which implies a = 0. Since the spectrum of a bounded operator is compact, the

spectral measure µ of J is supported in a compact set Σ,

m(z) =

(e1,

1

J0 − zIe1

)= b−

∫Σ

λ

λ2 + 1dρ(λ) +

∫Σ

1

λ− zdρ(λ)



where ∫Σ

1

λ− zdρ(λ) = −z−1

∫Σ

dρ(λ) +O(z−2)

as z →∞, from which we conclude that b−∫

Σ

λ

λ2 + 1dρ(λ) = 0 and

m(z) =

∫ ∞−∞

1

λ− zdρ(λ) . (B.8)

is a Borel transform of the spectral measure µ = dρ.

The Weyl m–function can be approached, as originally has been done by Weyl,

as a sequence limit: m(z) = limN→∞mN (z) of holomorphic functions in C\σ(J0,β).

Here, J0,β is a finite Jacobi matrix satisfying 0–Dirichlet boundary condition at 0

and β–boundary condition at N :

uN cosβ − uN+1 sinβ = 0 . (B.9)

A solution χ(z) = χ(z;N) of Schrodinger equation (B.1b),

χ(z) := w(z)−mN (z)y(z)

satisfies the phase β–boundary condition (B.9) at N iff

mN (z) =wN (z)− ζwN+1(z)

yN (z)− ζyN+1(z), ζ = tanβ . (B.10)

Since the r.h.s. of (B.10) is a linear fractional map L : C −→ C, L =

(aζ + b) / (cζ + d) with ad− bc 6= 0, as ζ varies over R, mN (z), for fixed N ∈ N and

z ∈ C with =z 6= 0, varies over a circle KN in C, called Weyl circle. The fact that

χ(z) satisfies (B.9) may also be expressed as ζ = χN (z)/χN+1(z).

Proposition B.4. The inequality

‖χ(z)‖2N ≤=mN (z)

=z(B.11)

is precisely the condition for mN to lie inside a Weyl circle KN :

|cN −m|2 = r2N (B.12)

of center cN = WN [w, y]/WN [y, y] and radius rN = |WN [y, y]|−1.

Proof. Let us show that (B.12) is the Weyl circle. We write the circle equation

=ζ = =(χN/χN+1) = 0 as

0 = WN [χ, χ] = WN [v, v]−mNWN [y, v]−mNWN [v, y]+ |mN |2WN [y, y] . (B.13a)

Together with (B.3d), WN [g, h] = −WN [h, g] = −WN [h, g] and

|WN [g, h]|2 = WN [g, h]WN [g, h] +WN [g, g]WN [h, h] (B.13b)

with g = y and h = w, (B.13a) multiplied by 1/WN [y, y] (WN [y, y] 6= 0, by (B.13e)

below) reads

− 1

|WN [y, y]|2+WN [y, w]

WN [y, y]

WN [y, w]

WN [y, y]−mN

WN [y, w]

WN [y, y]− mN

WN [y, w]

WN [y, y]+ |mN |2 = 0



which is exactly −r2N + |cN −mN |2 = 0. Hence, by (B.13a), mN lies in KN iff

WN [χ, χ]

WN [y, y]= 0 . (B.13c)

Using again the fact that mN is a linear fractional map, either mN satisfies

|cN −mN | ≤ rN or it is outside the Weyl circle KN . Replacing mN in the circle

equation (B.13c) by its center cN , the l.h.s. of (B.13c) is the negative number −r2N .

So, mN is inside or at KN iffWN [χ, χ]

WN [y, y]≤ 0 . (B.13d)

Applying Green’s identity (B.3a) with w replaced by y, together with W0[y, y] = 0,

yield

WN [y, y] = −=z ‖y‖2N . (B.13e)

Similarly, with W0[χ, χ] = =mN , we have

WN [χ, χ] = =mN −=z ‖χ(z)‖2N . (B.13f)

Replacing (B.13e,f) into (B.13d), we conclude that mN is inside or at KN iff

=z ‖χ(z)‖2N −=mN

=z ‖y‖2N≤ 0

which is equivalent to (B.11).

q.e.d.

Problem B.12. Show (B.13b).

The radius rN = rN (z) =(=z ‖y(z)‖2N

)−1

is, by (B.13e) and definition (B.12),

a monotone decreasing function of N and the circles KN (z), N ≥ 1, contract to a

limit point by proposition B.2. It thus follows that mN (z) converges, as N → ∞,

to the Titchmarsh–Weyl m–function m(z), uniformly in compact sets of H, and

w(z)−m(z)y(z), =z > 0, is the (only one) l2(Z+)–solution of (B.1b).

Spectral decomposition in terms of the values of m(λ+ i0) The spectral

measure µ = dρ may be decomposed into absolutely continuous, singular, singular

continuous and pure point components: µ = µac + µs, µs = µsc + µpp, according to

the boundary value =m+(λ) = limε↓0=m(λ + iε) for which the Stieltjes inversion

formula (lemma 1, chapter 2 of [24])

ρ (λ+)− ρ (λ−) = limε↓0

1

π

∫ λ+

λ−

=m(λ+ iε)dλ

formula plays a fundamental role.

A function f(z) is said to possess a normal limit at point λ ∈ R if f(z)

converges to a (finite or infinite) limit as z ↓ λ along perpendicular to real axis

direction. Since m(z) is Herglotz, we have (see [17])

Lemma B.1. If the Radon–Nikodym derivative (dµ/dλ) (λ) of the spectral measure

µ at λ w.r.t. Lebesgue measure dλ exists, finite or infinitely, then =m+(λ) exists

and (dµ/dλ) (λ) = (1/π)=m+(λ).



Definition B.9. A set Σ ⊂ R is called minimal (or essential) support of a

measure ν in R if

i. ν(R/Σ) = 0 (i.e. Σ is the support of ν)

ii. Any subset Σ0 ⊂ Σ which does not support Σ has ν and Lebesgue measure 0:

ν(Σ0) = L(Σ0) = 0 .

It follows from de la Vallee–Poisson (theorem 9.6, chapter IV of [46]), Lebesgue–

Radon–Nikodym theorem (theorem 6.9 of [44]) and lemma B.1 that (see proposition

1 of [17])

Proposition B.5. The minimal supports Σ, Σac, Σs, Σsc and Σpp of µ, µac, µs,

µsc and µpp, the spectral measure µ of a Jacobi matrix J , and the absolutely con-

tinuous (ac), singular (s), singular continuous (sc) and pure point (pp) parts, are

respectively given by

i. Σ = λ ∈ E : 0 < =m+(λ) ≤ ∞ii. Σac = λ ∈ E : 0 < =m+(λ) <∞

iii. Σs = λ ∈ E : 0 < =m+(λ) =∞iv. Σsc = λ ∈ E : 0 < =m+(λ) =∞, L(λ) = 0v. Σpp = λ ∈ E : 0 < =m+(λ) =∞, L(λ) > 0

where E = λ ∈ R : =m+(λ) exists.

Now, we introduce a key notion in the Gilbert–Pearson theory relating solutions

of the Schrodinger equation (B.1b) and the decomposition of the spectral measure

µ of J .

Definition B.10. A solution u = (un)n≥0 of(Jφ − λI

)u = 0 (regardless the phase

boundary φ) is said to be subordinate iff

limL→∞

‖u‖L‖w‖L

= 0 (B.14)

holds for any linearly independent solution w = (wn)n≥0 of the equation, where

‖u‖2L =

L∑n=1

|un|2 denotes the norm over an interval of length L.

Remark B.2. Gilbert–Pearson’s theory deals with boundary values of the Weyl–

Titchmarsh m–function. Theorems 1 and 2 of [17] (see [29] for the discrete case)

prove that m+(λ) = limε↓0m(λ + iε) exists if, and only if, a subordinate solution

of (J − λI)u exists which (when it exists) is given by χ(λ) = w(λ)−m+(λ)y(λ) if

|m+(λ)| <∞ and by y(λ) if |m+(λ)| =∞. By (B.7), if m+(λ) exists, then m+α (λ)

exists for every α ∈ (0, π); if it exists and |m+(λ)| = ∞, then m+α (λ) is finite for

every α ∈ (0, π); if m+(λ) exists and is finite, then there is only one α ∈ (0, π) such

that |m+α (λ)| =∞.



It turns out that the absolutely continuous part dρac of spectral measure dρ is

supported on a set Σac of λ’s for which no subordinate solution of(Jφ − λI

)u exists

and the singular part dρs is supported on a set Σs of λ’s for which u obey the phase

boundary φ and the Radon–Nikodym derivative dρ/dλ = (1/π)=m+φ (λ) diverges.

More precisely, we have

Proposition B.6. Let Jφ be an admissible operator in l2(Z+) satisfying φ–phase

b.c. (B.1c) and whose Schodinger equation is (B.1b). Let µ = dρ be its spectral

measure (B.8). Then, the minimal supports Σ, Σac, Σs, Σsc and Σpp of µ, µac, µs,

µsc and µpp are respectively given by

i. Σ = R/Σ0 where Σ0 =

λ ∈ R : there exists a subordinate solution but it does not satisfy (B.9)ii. Σac = λ ∈ R : no subordinate solution exists

iii. Σs = λ ∈ R : there exists a subordinate solution that satisfies φ-phase b.c.iv. Σsc =

λ ∈ R : there exists a subordinate solution satisfying φ-phase b.c. but does not belong to l2(Z+)v. Σpp =

λ ∈ R : there exists a l2(Z+) subordinate solution that satisfies φ-phase b.c..

Transfer matrix We restrict ourselves to the two sparse models we have dealt

with. For our convenience, we consider off–diagonal (??)-(??) analogous version of

the model Jω = J0 + V ω defined by (??-c) but the formulas derived for the latter

can be translated to the former without difficulties.

Definition B.11. Given λ ∈ R and n ∈ N, a (n–step) transfer matrix T (n;λ) is

a product

T (n;λ) := T (n, n− 1;λ)T (n− 1, n− 2;λ) · · ·T (1, 0;λ) (B.15)

of the (one–step) 2× 2 transfer matrix

T (k, k − 1;λ) =1

pk

(λ −pk−1

pk 0

)(B.16)

(replaced by

(λ− vk −1

1 0

)for the diagonal model (??-c)).

Any solution u = (un)n≥0 of the Schrodinger equation((Jφ − λI

)u)n

= pnun+1 + pn−1un−1 − λun = 0 (B.17)

for n ≥ 1, p0 ≡ 1, with φ–phase boundary condition at 0 (B.1c) satisfies(uk+1

uk

)= T (k, k − 1;λ)

(ukuk−1

). (B.18)

We denote by uφ =(uφn)n≥0

, the solution of (B.17) “normalized” by u20 + u2

1 = 1.



Proposition B.7. Let (pn)n≥0 be a sparse sequence:

pn =

p if n = aj ∈ A1 otherwise

with 0 < p < 1 and A defined by (??). Then, there exists a real 2× 2 matrix U so

that the n–step transfer matrix conjugated by U reads

UT (n;λ)U−1 = R ((n− aj)ϕ)P+−R (βjϕ) · · ·P+−R (β1ϕ) , (B.19)

for aj < n < aj+1 and λ = 2 cosϕ, ϕ ∈ (0, π), where

R(θ) =

(cos θ sin θ

− sin θ cos θ

)(B.20)

and

P+− =

(p 0

(1/p− p) cotϕ 1/p

). (B.21)

Proof. Let λ = 2 cosϕ and let

U =

(0 sinϕ

1 − cosϕ

)(B.22)

be a 2 × 2 nonsingular matrix that conjugates the free transfer matrix T0(λ) =(λ −1

1 0

)into a clockwise rotation matrix:

Problem B.13. Show that UT0(λ)U−1 = R(ϕ).

(B.19) follows by (B.15), (B.20) and P+− = R (−ϕ)UT (aj + 1, aj ;λ)T (aj , aj −1;λ)U−1R (−ϕ) for each aj ∈ A.

q.e.d.

Prufer variables Definition B.12. Let uφn and U be as in definition B.11 and

(B.22). The real valued functions Rj = Rj(ϕ, φ) and θj = θj(ϕ, φ), j = 0, 1, 2, . . .,

given by

Rj

(cos θjsin θj

):= U

(uφaj+1

uφaj

), (B.23)

are called Prufer variables.

Proposition B.8 ( [37], pp. 770, 771 and 776). Let (pn)n≥1 be as in Proposi-

tion B.7. Then,

θj = g(ϕ, θj−1)− (βj + ωj − ωj−1)ϕ (B.24a)

1

R2j

= F (ϕ, θj)1

R2j−1

(B.24b)



hold for any j ∈ N with (R0, θ0) ∈ R+ × [0, π], βj given by (??), ωjj≥1 i.i.d.

random variables,

g(ϕ, θ) = tan−1

(1

p2(tan θ + cotϕ)− cotϕ

), (B.24c)

F (ϕ, θ) =p2

a+ b cos 2θ + c sin 2θ, (B.24d)

and a, b and c are functions of p and ϕ:

2a =(1− p2

)2cot2 ϕ+ 1 + p4

2b =(1− p2

)2cot2 ϕ− 1 + p4

c =(1− p2

)cotϕ .

Proof. It follows directly from (B.19) and (B.23).

q.e.d.

Problem B.14. Prove the recursive relations (B.24b,b).

An explicit computation yields

a+ b cos 2θ + c sin 2θ ≥ minθ

(a+ b cos 2θ + c sin 2θ) = a−√a2 − p4 > 0 , (B.25)

and implies that F (ϕ, θ) is uniformly bounded in [0, π]× [0, π] and has unit mean:

F :=1

π

∫ π

0

F (ϕ, θ)dθ =p2

√a2 − b2 − c2

= 1 (B.26)

by equations (26) and (29) of [41] together with b2 + c2 = a2 − p4.

Write

v0 ≡ R0

(cos θ0

sin θ0

):= U

(cosφ

sinφ

), (B.27)

i.e., R0 =√

1− sin 2φ cosϕ and tan θ0 = cotφ/ sinϕ − cosϕ with 0 < θ0 < π. For

n and j such that aN < n < aN+1, the Euclidean norm∥∥UT (n;λ)U−1v0

∥∥2= R2

N , (B.28)

can be written as

R2N =

R2N

R2N−1

· · · R21

R20

R20 = exp

(N∑k=1

f (ϕ, θk)

)R2

0 , (B.29)

where

f(ϕ, θ) = − logF (ϕ, θ) (B.30)

by (B.24b), (B.24d) and (B.25), is uniformly bounded in [a, b]×[0, π], 0 < a < b < π,

and has mean

f =1

π

∫ π

0

f(ϕ, θ)dθ = log

(a+√a2 − b2 − c2

2p2

). (B.31)



The quantity between parenthesis r = r(p, ϕ) =a+√a2 − b2 − c2

2p2, is explicitly

given by

r = 1 +v2

4csc2 ϕ

with v = (1− p2)/p or, using λ = 2 cosϕ, by

r = 1 +v2

4− λ2, (B.32)

and coincides with the analogous expression for the diagonal model (??-c). We shall

employ it in the following for both diagonal and off–diagonal sparse models.

Spectral nature criteria We gather a collection of results on spectral analysis

useful for the sparse models in consideration. We begin with the following (lemma

3.1 of [33])

Proposition B.9. Let uφ and T (n;λ) be defined as in definition B.11. Then∥∥uφ+π/2∥∥2

L+1

‖uφ‖2L+1

≤ 2

(1

L

L∑n=1

‖T (n;λ)‖2)2

. (B.33)

Proof. By definition B.11, uφ and uφ+π/2 satisfy the initial conditions (B.2) with

α = φ and it thus follows that

Tφ(n;λ) := T (n;λ)R (−φ) =

(uφn+1 u

φ+π/2n+1

uφn uφ+π/2n

)(B.34a)

and

1 = Wn

[uφ, uφ+π/2

]= −pn detTφ(n;λ), (B.34b)

by (B.18), (B.20), (B.3b) and (B.3c). Let the first and second columns of (B.34a)

be denoted by uφn and vφn. In terms of these vectors, the Wronskian reads

Wn

[uφ, uφ+π/2

]= pnuφn ·J vφn, with J =

(0 1

−1 0

)and by (B.34b), Cauchy-Schwarz

inequality, pn ≤ 1 and ‖J ‖ = 1∥∥uφn∥∥ ∥∥vφn∥∥ ≥ pn ∣∣uφn · J vφn∣∣ = 1 . (B.34c)

By definition (B.34a)

1

L

L∑n=1

∥∥vφn∥∥2 ≤ 1

L

L∑n=1

‖T (n;λ)‖2

holds (term–by–term) and, by ,

1 ≤

(1

L

L∑n=1

∥∥uφn∥∥∥∥vφn∥∥)2

≤ 1

L

L∑n=1

∥∥uφn∥∥2 1

L

L∑n=1

∥∥vφn∥∥2



by (B.34c) and Cauchy-Schwarz inequality. From the last two equations, one con-

cludes ∑Ln=1

∥∥vφn∥∥2∑Ln=1

∥∥∥uφn∥∥∥2 ≤

(1

L

L∑n=1

‖T (n;λ)‖2)2

. (B.34d)

and this, together with

L+1∑n=1

∣∣∣uφ+π/2n

∣∣∣2 ≤ L∑n=1

∥∥vφn∥∥2

L+1∑n=1

∣∣uφn∣∣2 ≥ 1

2

L∑n=1

∥∥uφn∥∥2, (B.34e)

yields (B.33).

q.e.d.

Let µ = dρ be the spectral measure of Jφ:(e1, f

(Jφ)e1

)=

∫f(λ)dρ(λ) (B.35)

for every bounded measurable function with compact support in (−2, 2). The vector

e1 is cyclic for Jφ in the sense that(Jφ)ke1 : k ∈ N

is a dense set in the Hilbert

space H and any other spectral measure µΨ is absolutely continuous with respect

to µ. According to proposition B.6,

Σac = λ ∈ [−2, 2] : there is no subordinate solution (B.36)

is an essential support of µac and has zero measure with respect to the singular part

µs. Proposition B.9 gives one–half of the following

Theorem B.1 (Theorem 1.1 of [33]). The essential support Σac of the a.c. part

µac is given by

Σac =

λ : lim inf

L→∞

1

L

L−1∑n=0

‖T (n;λ)‖2 <∞

. (B.37)

Proof. Let us denote by S the r.h.s. of (B.37). If Jφ has a subordinate solution

at λ then the r.h.s. of (B.33) diverge. So, S is contained into Σac and we quote

proposition 3.3 of [33] for a proof that it is an essential support of dρac, i.e., for

almost every λ ∈ [−2, 2] with respect to µac, we have λ ∈ S.

q.e.d.

Now, we come back to equations (B.28)–(B.32). Suppose that f = f(ϕ, θ) in

(B.29) is replaced by its average f = f(ϕ) given by (B.31) and

EN (ϕ) =1

N

N∑k=1

(f (ϕ, θk)− f(ϕ)

)−→ 0 , as N →∞ for a.e. ϕ ∈ [0, π) .

(B.38)



Since by definition (B.32), r > 1 if v > 0 and by (B.28)

‖T (n;λ)‖2 ∼ rj

for aj < n < aj+1 a.e. λ ∈ (−2, 2) (see equations (B.42a) and (B.42c) below and

(??), for a derivation), we conclude by theorem B.1 the following

Theorem B.2. Let µ = dρ be the spectral measure of Jφ and suppose that the

hypothesis (B.38) holds. Then the essential support of its a.c. part µac, Σac = ∅.

The conclusions is not affected by the exclusion of a set A ⊂ [−2, 2] of Lebesgue

measure zero, by definition B.9 of essential support Σac. The presence of randomness

ω on the sparse models in consideration allows the hypothesis to be rigorously proved

(theorem ??).

Metric properties of spectral measure We state without proof an extension

to Gilbert-Pearson theory, due Jitomirskaya–Last [23], which relates the Hausdorff

decomposition of the spectral measure to a generalized subordinacy solution:

Theorem B.3 (Theorem 1.2 of [23]). Let µ = dρ be given by (B.35), λ ∈ [−2, 2]

and α ∈ (0, 1). Then, the upper Hausdorff derivative

Dαµ(λ) := lim sup

ε→0

ρ ((λ− ε, λ+ ε))

(2ε)α =∞

iff

lim infL→∞

∥∥uφ∥∥L∥∥uφ+π/2

∥∥α/(2−α)

L

= 0 .

One consequence of Theorem B.3 (see Corollary 4.4 of [23]) may be restated in

terms of the transfer matrix T (n;λ).

Corollary B.1 (Corollary 3.7 of [10]). Suppose that for some α ∈ [0, 1) and ev-

ery λ in a Borel set A,

lim supl→∞

1

l2−α

l∑n=0

‖T (n;λ)‖2 <∞ . (B.39)

Then, the restriction ρ(A ∩ ·) of ρ to the set A is α–continuous.

The other consequence is (see Corollary 4.5 of [23])

Corollary B.2 ((Corollary 3.8 of [10])). Suppose that

lim infl→∞

∥∥uφ∥∥2

l

lα= 0

holds for every λ in some Borel set A. Then, the restriction ρ(A∩ ·) is α–singular.



The proof of both corollaries relates the behavior of the eigenvectors uφn and

uφ+π/2n with the norm of T (n;λ) and the Prufer radius Rk. Note that

R2j (φ) =

∥∥∥∥UT (n;λ)

(cosφ

sinφ

)∥∥∥∥2

=(uφn+1

)2

+(uφn)2 − 2uφn+1u

φn cosϕ (B.40a)

holds with aj < n < aj+1, φ ∈ (0, π) and λ = 2 cosϕ ∈ (−2, 2), by (B.23), (B.34a),

(B.22) and (B.20). From (B.40a) we deduce

(1− |cosϕ|)((

uφn+1

)2

+(uφn)2) ≤ R2

j (φ) ≤ (1 + |cosϕ|)((

uφn+1

)2

+(uφn)2)

(B.40b)

Problem B.15. Let ‖A‖U :=∥∥UAU−1

∥∥ be a matrix norm defined by (B.22) and

‖B‖ = supv∈C2 ‖Bv‖ / ‖v‖. Show that

C−1 ‖A‖ ≤ ‖A‖U ≤ C ‖A‖ (B.41)

holds with C =√

(1 + |cosϕ|) / (1− |cosϕ|).

Problem B.15 together with (B.28), imply

‖T (n;λ)‖ ≥ C−1 ‖v0‖−1 ∥∥UT (n;λ)U−1v0

∥∥ = C ′Rj(φ) (B.42a)

and

‖T (n;λ)‖ ≤ C supv:‖v‖=1

∥∥UT (n;λ)U−1v∥∥ = C ′′Rj(φ) . (B.42b)

where v =(cos θ, sin θ

)is the (unique) unit vector for which the supremum is

attained and φ solves θ0(φ) = θ for φ. To get rid of the sup, an artifact introduced

in theorem 2.3 of [31] can be used (see (B.51d)) to replace equation (B.42b) by

‖T (n;λ)‖ ≤ C max(Rj(φ

1), Rj(φ2))

(B.42c)

with C = C/∣∣sin (θ1

0 − θ20

)/2∣∣ and θi0 = θ0(φi), i ∈ 1, 2. See quoted reference for

completion of the proof.

Proposition B.10. If (βj)j≥1 satisfies a super–exponential sparseness condition

limj→∞ βj−1/βj = 0, then the spectrum σ(Jφ) of Jφ has Hausdorff dimension 1.

Proof. It follows from (B.42c) and (??) that, if (βj)j≥1 satisfies a super–exponential

sparseness, then for any Borel set A ⊂ [−2, 2] and ε > 0 there exist n0 = n0(ε,A)

such that

‖T (n;λ)‖2 ≤ Cjrj ≤ nε

holds for n ≥ n0, uniformly in A. So,

1

l2−α

l∑n=1

‖T (n;λ)‖2 ≤ Clα−1+ε



holds for l sufficiently large and the lim-sup is finite for any α > 1. This, to-

gether with Corollary B.1 and definition of Hausdorff dimension (see definition ??),

concludes the proof.

q.e.d.

We turn now to the point spectrum, which matters only for sequence (βj)j≥1

exponentially sparse

βj = βj (B.43)

for some integer β > 1, asymptotically as j →∞. By Theorems 1.6 and 1.7 of [33](Jφ − λI

)u = 0 has no l2 (N)–solutions if

∞∑n=1

‖T (n;λ)‖−2=∞ (B.44)

(see also Theorem 2.1 in [48]) and an l2 (N)–solution if

∞∑n=1

‖T (n;λ)‖2( ∞∑k=n

‖T (k;λ)‖−2

)2

<∞ . (B.45)

Condition (B.45) can be optimized for sparse models by improving Theorems

8.1 and 8.2 of [33] on the existence of subordinate and l2 (N)–solutions. By equation

(B.19), for n and N s.t. aN < n < aN+1, we have

‖T (n, λ)‖U ≤N∏k=1

‖P+−R (βkϕ)‖ ≤ ‖P+−‖N .

This together with (B.41) imply that ‖T (n;λ)‖ cannot growth exponen-

tially fast in n. As a consequence, the largest Lyapunov exponent, γ =

limn→∞(1/n) log ‖T (n;λ)‖, vanishes for T (n;λ) as a product of one-step trans-

fer matrices but it is strictly positive as a product of the P+−R (βkϕ):

limN→∞(1/N) log ‖P+−R (βNϕ) · · ·P+−R (β1ϕ)‖ > 0 a.e. ϕ.

We follow closely Lemma 4.1 of [11] and Proposition 3.9 of [10], where decay at

infinity of a subordinate solution were established (see also lemma 2.1 of [55]).

Proposition B.11. Let Jφ be given by (B.17) with sparse sequence (βn)n≥1 and

let tn denote

‖T (an + 1;λ)‖. Suppose that∞∑n=1

t−2n <∞ (B.46)

holds for some λ ∈ [−2, 2]. Then, there exist a unit vector v∗ = (cosφ∗, sinφ∗) with

φ∗ ∈ [0, π) so that

‖T (n;λ) v∗‖2 ≤ B

( ∞∑M=N

t−2M

)2

t2N + t−2N (B.47)

is satisfied for some constant B <∞ and every aN < n < aN+1 with N ∈ N.



Corollary B.3. Under the assumptions of Proposition B.11, the vector uφ∗

n :=

T (n;λ) v∗ =

(uφ∗

n+1

uφ∗

n

)defines a strong subordinate solution uφ

∗=(uφ∗

n

)n≥0

in

the sense that

limn→∞

∥∥uφ∗n ∥∥∥∥∥uφn∥∥∥ = 0

holds for any φ ∈ [0, π) with φ 6= φ∗.

Corollary B.4. The strong subordinate solution uφ∗

=(uφ∗

n

)n≥0

is an l2 (Z+)–

solution of(Jφ − λI

)u = 0 with φ = φ∗, provided

∞∑n=1

βnt−2n <∞ and

∞∑n=1

( ∞∑m=n

t−2m

)2

βnt2n <∞ (B.48)

are verified in addition to the assumptions of proposition B.11.

Corollary B.5. Suppose, in addition to the assumptions of proposition B.11, that

C−1n rn ≤ t2n ≤ Cnrn (B.49)

holds with r > 1 and C1/nn 1 as n tends to ∞. Then∥∥∥uφ∗n ∥∥∥2

≤ Cnr−n (B.50)

holds with C1/nn 1 as n tends to ∞.

Remark B.3. As the Prufer angles (θωn)n≥1 are uniformly distributed mod π (the-

orem ??), equation (B.49) actually holds with r given by (B.32) a.e. λ. Theorem

?? excludes a set of λ’s which is countable and independent of the initial Prufer

angle θ0. These properties are crucial for proving pure point spectrum. Note that

(B.42a) and (B.42a) hold for some θi0, i = 1, 2, where θ0, by (B.27), depends on λ

and on the phase boundary φ.

Proof of proposition B.11. Let λ = 2 cosϕ and write

T (an + 1;λ) = An(λ) · · ·A1(λ)

where

UAkU−1 = P+−R (βkϕ) .

Let sn = ‖An(λ)‖ denote the spectral norm of An(λ). It follows from (B.15)

sn ≤ C ‖An(λ)‖U ≤ C ‖P+−‖ ≤ D (B.51a)

with D = C

√1 + (p− 1/p)

2csc2 ϕ, by (B.21), uniformly in n. As a consequence,

∞∑n=1

s2n+1

t2n≤ D2

∞∑n=1

1

t2n<∞ , (B.51b)



by hypothesis (B.46), verifies the assumption of Theorem 8.1 of [33].

The transfer matrices Ak’s are 2×2 unimodular real matrices. Since unimodular

matrices form an algebra, the product of T (an + 1;λ) with its adjoint T ∗(an +

1;λ) is a 2 × 2 unimodular symmetric real matrix with eigenvalues t2n and t−2n

and corresponding (orthonormal) eigenvectors v+n and v−n : v+

n · v−n = 0. Write

vφ =

(cosφ

sinφ

)and define φn by

vφn = v−n . (B.51c)

Clearly, v+n = vφn+π/2 and

‖T (an + 1;λ)vφ‖2 = vφ · T ∗(an + 1;λ)T (an + 1;λ)vφ

= t2n∣∣vφ · v+

n

∣∣2 + t−2n

∣∣vφ · v−n ∣∣2= t2n sin2 (φ− φn) + t−2

n cos2 (φ− φn) . (B.51d)

by the spectral theorem.

The completion now follows exactly the steps of the proof of Theorem 8.1 of

[33]. The conclusion of proposition B.11, equation (B.47), is the combination of

equations (8.5) and (8.7) of [33]. We review the main steps. Firstly, the sequence

(φn)n≥1 converges to φ∗ ∈ [0, π] under the condition (B.51b). The key estimate

|φn − φn+1| ≤π

2

s2n+1

t2n, (B.51e)

established using properties of a matrix norm, (B.51d) for n + 1 and (B.51c), to-

gether with (B.51b) implies that (φn)n≥1 is a Cauchy sequence. Next, the telescope

estimate

|φn − φ∗| ≤∞∑m=n

|φm − φm+1| ≤π

2

∞∑m=n

s2m+1

t2m

replaced into (B.51d) yields

‖T (an + 1;λ)vφ∗‖2 ≤ t2n (φ∗ − φn)2

+ t−2n

≤ Bt2n

( ∞∑m=n

t−2m

)2

+ t−2n (B.51f)

with B =π

2D4, which concludes the proof.

q.e.d.

Proof of Corollary B.3. Let v∗ = vφ∗ and v∗ = vφ∗+π/2. By (B.51d),

‖T (an + 1;λ)v∗‖2 ≥ t2n/2 is satisfied for sufficiently large n and for any φ 6= φ∗

we have vφ = av∗ + bv∗, with a2 + b2 = 1, a 6= 0, and ‖T (an + 1;λ)vφ‖2 ≥ a2t2n/2

holds for some n sufficiently large. By (B.51d),∥∥uφ∗n ∥∥2∥∥∥uφn∥∥∥2 ≤2

a2

B( ∞∑m=n

t−2m

)2

+ t−4n

−→ 0



by (B.46), concluding the proof.

q.e.d.

Proof of Corollary B.4. We prove that the strongly subordinate solution is a l2(Z+)–

solution under assumptions (B.46) and (B.48). For any k ∈ N let n be such that

an + 1 ≤ k < an+1 holds. By (B.41),

‖T (k;λ)vφ∗‖2U ≤ C ‖vφ∗‖2

which, by (B.43), (B.40a) and (B.40b), together with (B.47), yields

∞∑k=1

∥∥∥uφ∗k ∥∥∥2

≤ 1

2(1− |cosϕ|)

∞∑k=1

‖UT (k;λ)vφ∗‖2

≤ B′∞∑n=1

βnt2n

( ∞∑m=n

t−2m

)2

+B′′∞∑n=1

βnt−2n

for some constants B′ and B′′, concluding the proof.

q.e.d.

Proof of Corollary B.5. Equation (B.50) is a direct consequence of (B.47) and

(B.49).

q.e.d.


Appendix C

Symmetric Cantor sets and relatedsubjects

Let AB be a closed segment of length l and let

0 < ξ < 1/2 (C.1)

be a number (the “ratio”). Following [28], consider a trisection of the segment AB

in parts respectively equal to lξ, l(1 − 2ξ), and lξ, and remove the open central

interval (“black” interval) of length l(1 − 2ξ); there remain two closed intervals

(“white” intervals) of common length lξ. Such a dissection of the given interval

AB will be said to be a dissection of type (2, ξ), the number 2 recalling that after

dissection there remain two white intervals.

We now start from a fundamental interval [a, b] (often the interval [0, 2π]). Let

us operate a dissection of type (2, ξ1) on this interval, then one of type (2, ξ2) over

each of the two remaining white intervals; further, a dissection of type (2, ξ3) over

each of the 22 white intervals obtained, and so on, the infinite sequence ξk∞1 being

such that

0 < ξk < 1/2 , for all k = 1, 2, . . . (C.2)

At the k–th step, we shall have a set Ek consisting of 2k white intervals of common

length

(b− a)ξ1 · · · ξk . (C.3)

The intersection

E ≡∞⋂k=1

Ek (C.4)

is a perfect set: E is closed and every point of E is a limit point of Ek. Further,

it has no interior points, i.e., its closure is nowhere dense; it is a Cantor set. Its

contiguous points are the edges of the black intervals obtained in the course of all

the dissections. We have (L will denote Lebesgue measure in this appendix), by

(C.3):

L(E) = 0⇔ limk→∞

2kξ1 · · · ξk = 0 (C.5)

41



Special cases of the symmetric Cantor sets described above are prototypes of

self-similar sets. In the case ξk = ξ < 1/2 above one has a homogeneous perfect

set E of type (2, ξ), but this notion may be generalized, see [28], p. 16. This set

E may be written as E = ∪2i=1fi(E), where f1(x) = x/3 and f2(x) = x/3 + 2/3.

The generalization of this structure to T = ∪Ni=1fi(T ), where T is a nonempty

compact set and (f1, . . . , fN ) are similitudes, i.e., functions of type fi(x) = λix+bi,

with 0 < λi < 1 is called an iterated function system [16](IFS) for T , and

T the attractor or the invariant set for the IFS. If there exists an open set V

such that fi(V ) ⊂ V and fi(V ) ∩ fj(V ) = ∅ for i 6= j, we say that the IFS

satisfies the open set condition. This is the case for the Cantor set of type (2, ξ),

with V = (0, ξ). Even further generalizations to Rn exist: see [16]. Measures

supported by these sets are called self-similar measures, and their construction

has been described in detail by Guarneri and Schulz-Baldes in a beautiful review

[18]: the latter authors also describe the construction of Schrodinger operators (one-

dimensional Jacobi matrices) having the previously mentioned self-similar measures

as spectral measures µΨ with Ψ = δ0 as cyclic vector (see theorem ??). This

construction was used in models of quasicrystals in three dimensions, see chapter

?? section ??.

Claim C.1. The 2k origins of the white intervals of which Ek consists are given by

the basic formula

a+ l[ε1(1− ξ1) + ε2ξ1(1− ξ2) + · · ·+ εkξ1 · · · ξk−1(1− ξk)] (C.6)

with εi ∈ 0, 1 ∀i = 1, . . . , k and l = b− a.

Proof. By induction: for k = 1, a+ l[ε1(1− ξ1)], for ε1 ∈ 0, 1 is s.t.

a+ 0 = a and a+ l(1− ξ1) = a+ lξ1 + l(1− 2ξ1)

Suppose, now, k ≥ k0 ≥ 1, and that the 2k0 origins are given by

a+ l[ε1(1− ξ1) + · · ·+ εk0ξ1 · · · ξk0−1(1− ξk0)] (C.7a)

For k = k0 + 1, one has, to each (C.7a), to add lξ1 · · · ξk0(1 − ξk0+1), but this

corresponds to build

a+ l[ε1(1− ξ1) + · · ·+ εk0ξ1 · · · ξk0−1(1− ξk0) + εk0+1ξ1 · · · ξk0(1− ξk0+1)] (C.7b)

with εk0+1 ∈ 0, 1. This ends the induction.

q.e.d.

Since limk→∞ ξ1 · · · ξk = 0 (because of (C.2)), the points of E given by (C.4) are

given by the infinite series

x = a+ l[ε1(1− ξ1 + ε2ξ1(1− ξ2) + · · ·+ εkξ1 · · · ξk−1(1− ξk) + · · · ] (C.8)

with εk ∈ 0, 1 for all integers k ≥ 1. Define

rk = ξ1 · · · ξk−1(1− ξk) (C.9)


Symmetric Cantor sets and related subjects 43

Then, by (C.2),

rk > rk+1 + rk+2 + · · · (C.10)

We associate to each perfect symmetric set E the Lebesgue function which is

constructed in the following way. Let Lk(x) be the continuous function which

equals zero for x ≤ a, one for x ≥ b, increasing linearly by 12k

on each white interval

building Ek, and constant in each black interval contiguous to Ek. If k → ∞,

Lk(x) tends uniformly (see below) to a function L(x), continuous, constant in each

interval contiguous to E, monotone increasing, with L(a) = 0, L(b) = 1: the

Lebesgue function constructed on E. The same simple induction leading from

(C.7a) to (C.7b) shows that for all x ∈ E, where E is given by (C.4),

L(x) =ε12

+ε222

+ · · ·+ εk2k

+ · · · (C.11)

In order to see that the convergence is uniform, let Lp(x) =

p∑i=1

εi/2i; then

|Lp(x)− Lp+1(x)| ≤ 2−p; thus the limp→∞ Lp(x) is uniform.

The variation of L(x) along an interval will be called the L–measure of this

interval, denoted by dL; the Stieltjes integral of a continuous function f(x) w.r.t.

dL will be denoted by

∫ b

a

f(x)dL(x); in particular, the Fourier-Stieltjes transform

of dL,

Γ(u) ≡∫ ∞−∞

exp(iux)dL(x) =

∫ b

a

exp(iux)dL(x) (C.12)

Observing that L(x) grows by 1/2k on every white interval building up Ek, we have

Γ(u) = limk→∞

Γk(u) (C.13a)

where

Γk(u) ≡ 1

2k

∑εj∈0,1∀1≤j≤k

exp(iu[a+ l(ε1r1 + · · ·+ εkrk)]) (C.13b)

with rk defined by (C.9). Thus

Γk(u) = exp(iua)

k∏j=1

1 + exp(iulrj)

2

= exp

iua+ l/2

k∑j=1

rj

k∏j=1

cos(ulrj/2) (C.14a)

from which, using

∞∑j=1

rj = 1 and b− a = l,

Γ(u) = exp[iu(a+ b)/2]

∞∏j=1

cos(ulrj/2) . (C.14b)

Problem C.16. Prove the convergence of the infinite product on the r.h.s. of

(C.14b).



In particular, if [a, b] is centered at the origin, with a = −d, b = d,

Γ(u) =

∞∏j=1

cos(udrj) . (C.15)

Definition C.13. If all ξj have a common value ξ, E = Eξ is said to be a symmetric

Cantor set of constant ratio ξ.

Problem C.17. Prove that for Eξ a symmetric Cantor set of constant ratio ξ

constructed over [−d, d],

Γ(u) =

∞∏j=1

cos[ud(1− ξ)ξj−1] . (C.16)

The Hausdorff dimension dimH of the symmetric Cantor set of ratio ξ is

dimH(E) = | log 1/2| /| log ξ|(see [28]). There are several additional concepts of dimension which are relevant to

describe sets of this type, such as the information dimension or the fractal dimension

(see [15] for details). Let A be a set in Rn, and n(ε) the minimal number of balls

of radius ε which are necessary to cover A. The fractal dimension DF (A) is defined

by

DF (A) = limε→0

n(ε)

| log ε|For the set Eξ, considering the Ek = Ek,ξ in (C.4), take ε = ξ−k; then n(ε) = 2k,

and DF (E) = limk→∞log 2k

log ξk=

log 2

log ξ. For the set Eξ the Hausdorff and fractal

dimensions are equal. For any countable set, it is easy to see that both dimensions

yield zero, and for an interval in R, they yield one. Hence, a nonzero Hausdorff or

fractal dimension is a signal of uncountability (but there are uncountable sets of

zero Hausdorff measure, see chapter ?? for explicit examples). It is easy to see that

Eξ is uncountable; take, e.g., ξ = 1/3. The elements of E1/3 are all the numbers

which in base three representation

∞∑i=1

εi3−i have εi ∈ 0, 2 ∀i. We may, therefore,

define a one–to–one map of E1/3 to the set of numbers

∞∑i=1

εi2−i with εi ∈ 0, 1 ∀i,

i.e., to the whole interval [0, 1], hence the set E1/3 is uncountable.

We now show that the Lebesgue function L, given by (C.11), is UαH (see defi-

nition ??):

Proposition C.12. Let ξk = ξ for k = 1, 2, . . . with ξ satisfying (C.1). Then L,

given by (C.11), is UαH , i.e.,

L(x)− L(x′) ≤ C(x− x′)| log 1/2|| log ξ| (C.17a)

where, for definiteness,

0 < x− x′ < 1 (C.17b)



Proof. Suppose first that both x and x′ satisfy (C.17b) and are of the form (C.8),

then

x− x′ = l[(ε′p+1 − εp+1)ξp(1− ξ) + (ε′p+2 − εp+2)ξp+1(1− ξ) + · · · ] (C.18)

If the first p εi are equal for x and x′, for definiteness, then ε′p+1 − εp+1 = 1, and∣∣∣ε′p+k − εp+k∣∣∣ ≤ 1. Thus, from (C.18),

x′ − x ≥ lξp(1− ξ)− l[ξp+1(1− ξ) + ξp+2(1− ξ) + · · · ]= lξp(1− ξ)− lξp+1 = lξp(1− 2ξ) = Aξp (C.19a)

where, by (C.1),

0 < A <∞ , with A independent of x and x′ (C.19b)

By (C.11),

L(x′)− L(x) ≤ 2−p + 2−(p+1) + · · · = 2−p

1− 1/2= 22−p (C.19c)

By (C.19a), (x − x′)−1 ≤ A−1ξ−p, whence log(x − x′)−1 ≤ logA−1 + p| log ξ|, and

therefore

p ≥ − log(x− x′)| log ξ|

+logA

| log ξ|(C.19d)

By (C.19c-d),

L(x′)− L(x) ≤ C2− log(x′−x)| log ξ| = C(x− x)α , α ≡ | log 1/2|

| log ξ|(C.19e)

which is (C.17a). If x or x′ or both are not of the form (C.8), let x0 be the right

end point of the black interval containing x, and x′0 the left hand end point of the

black interval containing x′. As x−x′ > x0−x′0, and L(x′)−L(x) = L(x0)−L(x′0)

by the construction of L, (C.17a) subsists.

q.e.d.

We have now

Proposition C.13. Let Jα(L) be defined by (??) (for µ = L). Then, for any

0 < ε < α,

0 < Jα−ε(L) <∞ (C.20)

Proof. By proposition (C.1), L is UαH, and then, by Lemma ?? , (C.20) holds.

q.e.d.

Proposition C.14. Let d = π, and

ξ = 1/p (C.21a)

with p any odd integer such that

p ≥ 3 . (C.21b)

Then (C.20) holds, with α given by (C.19e), but

Γ(u) 9 0 , as u→∞ (C.21c)



Proof. By (C.16) with d = π,

Γ(u) =

∞∏j=1

cos[πu(1− 1/p)p−(j−1)] =

∞∏j=1

cos

[πu(p− 1)

pj

](C.21d)

Since p is an odd integer ≥ 3, it follows from (C.21d) that

Γ(n) = Γ(pn) ∀n ∈ Z (C.21e)

and thus (C.21c) is proved.

q.e.d.

Proposition C.14 shows that there exist singular measures which are not Rajch-

man measures (see chapter ?? section ??) and propositions C.13 and C.14 together

show that, also in one dimension, finiteness of the integral Jα−ε(µ) does not im-

ply pointwise convergence (see also chapter ?? section ??). The further problem

of whether it could happen that Γ(u) → 0 as u → ∞ except for u in a set of

zero Lebesgue measure, apparently left open by the proof of proposition C.14, is

negatively answered in section ?? (see Proposition ??).

The amusing remark (C.21e) goes back at least to [28]. Lyons [35] also mentions

it and remarks that, to our benefit, (C.21e) was not observed by F. Riesz, who,

instead, constructed his famous Riesz products [43] for precisely the purpose of

providing an example of a continuous measure which is not a Rajchman measure.

The proof of proposition C.12 is patterned after Salem’s paper [47].

We close this appendix with some important remarks. If, instead of (C.2)

1/2 < ξk < 1 (C.22)

the support of dL is the interval [−d, d]. By a theorem of Jessen and Wintner [22],

the corresponding measure may be either singular or a.c.; in the latter case, we have

an a.c. measure with nowhere dense support (see theorem ??). Such measures have

the following property [3]:

Proposition C.15. Suppose µ is an absolutely continuous measure supported by a

Cantor set C. Then µ /∈ L1(R).

Proof ([3], [8], chap. 10, p. 214) dµ(x) = f(x)dx for a function f supported by

C. Since C is a Cantor set, f cannot be continuous; but, if µ ∈ L1, f would be

continuous.

q.e.d.

Example C.1. The famous Vieta’s identity (A.1) may be rewritten as

sin t

t=

( ∞∏k=1

cos(t/22k−1)

)( ∞∏k=1

cos(t/22k)

)(C.23)

By ([27]), each product on the r.h.s. of (C.23) is the Fourier-Stieltjes transform

of a s.c. measure. Indeed, the second term on the r.h.s. of (C.23) is the Fourier-

Stieltjes transform µE of a measure supported by a perfect symmetric set E of



constant ratio ξ = 1/4 and is, therefore, s.c. ([27], p. 15). The first term on the

r.h.s. of (C.23) equals µE(2t) = µE′(t), where E′ is the perfect symmetric set of

constant ratio ξ = 1/4 but constructed on the interval [−π/2, π/2], which is, of

course, also s.c. by the same argument. Finally, the l.h.s. of (C.23) belongs to

L2(R,L) and it follows from theorem ?? (for p = 2) that µ is a.c..

Definition C.14 ([3]). Let H be a self–adjoint operator on a separable Hilbert

space H. The vector φ ∈ H is called a transient vector for H if

(φ, exp(−itH)φ) = O(t−N ) ∀N ∈ N (C.24)

The closure of the set of transient vectors is calledHtac (transient a.c. subspace).

One has [3, 8]:

Proposition C.16. Htac is a subspace of H such that Htac ⊂ Hac, and

Htac = φ : µφ ∈ L1(R) (C.25)

where the bar indicates closure. One defines

Hrac = H⊥tac ∩Hac (C.26)

which is called the recurrent a.c. subspace [3]: both Htac and Hrac are invariant

subspaces under H (see [3, 8]).

Chapter ?? section ?? discusses the connection of the above spectra with quan-

tum dynamical stability and provides or mentions explicit notrivial examples of

transient and recorrent a.c. spectra.


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