Spectral transition in a sparse model and a class of nonlinear dynamical systems

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Nonlinearity 20 (2007) 765–787 doi:10.1088/0951-7715/20/3/010

Spectral transition in a sparse model and a class ofnonlinear dynamical systems

Domingos H U Marchetti1, Walter F Wreszinski1, Leonardo F Guidi2 andRenato M Angelo3

1 Instituto de Fısica Universidade de Sao Paulo, Caixa Postal 66318, 05315 Sao Paulo, SP, Brasil2 Instituto de Matematica, Universidade Federal do Rio Grande do Sul, Av. Bento Goncalves,9500 - Predio 43-111, 91509 Porto Alegre, RS, Brasil3 Departamento de Fısica, Universidade Federal do Parana, Caixa Postal 19044, 81531 Curitiba,PR, Brasil

E-mail: marchett@if.usp.br, wreszins@fma.if.usp.br, guidi@mat.ufrgs.br andrenato@fisica.ufpr.br

Received 21 April 2006, in final form 11 January 2007Published 9 February 2007Online at stacks.iop.org/Non/20/765

Recommended by J R Dorfman

AbstractWe introduce a class of Jacobi matrices with sparse potential in the sensethat the perturbation of the Laplacian consists of a (direct) sum of fixed off-diagonal 2 × 2 matrices placed at sites whose distances from one another growexponentially. The essential spectrum is proved to be the set [−2, 2]. Themodel may be formulated in terms of angular variables, the Prufer angles,in terms of which is defined a nonlinear dynamical system with two controlparameters, p—the ‘impurity strength’—and β—the ‘sparseness’. A methodto study the spectrum of these matrices which exploits the existence, for fixedenergy, of a continuous asymptotic distribution function (a.d.f.) of the Pruferangles is introduced. This is in contrast to the method introduced by Pearson(1978 Commun. Math. Phys. 60 13–36), which exploits uniform distribution inenergy. With the help of metric theorems of ergodic theory and ideas of Zlatos(2004 J. Funct. Anal. 207 216–52), we are able to use this framework in orderto establish a singular continuous spectrum on an interval of positive Lebesguemeasure, under suitable conditions on β and p. In the complementary set ofparameters there is a (dense) pure point spectrum, and thus a spectral transitiontakes place, if a continuous a.d.f is assumed. We provide numerical evidencefor this assumption.

Mathematics Subject Classification: 47B36, 47A25, 47A35, 37N20, 37E05

0951-7715/07/030765+23$30.00 © 2007 IOP Publishing Ltd and London Mathematical Society Printed in the UK 765

766 D H U Marchetti et al

1. Introduction and summary

There have been a few results on models with spectral transition, i.e. supporting spectra ofdifferent types for a complementary set of parameters. They include the Anderson transitionin a Bethe lattice [K], a sort of metal–insulator transition for almost Mathieu operator [J], aswell as a transition from a singular continuous (s.c.) to a pure point (p.p.) spectrum in a specialrandom model with sparse potential [Z]. In this paper, we address the possibility of a spectraltransition in a deterministic model, a free Jacobi matrix perturbed by a sparse potential, similarto the one treated in [Z].

We consider off-diagonal Jacobi matrices

J =

0 p0 0 0 · · ·p0 0 p1 0 · · ·0 p1 0 p2 · · ·0 0 p2 0 · · ·...

......

.... . .

, (1.1)

with pn = pn(κ), n = 0, 1, 2, . . ., real-valued functions of a parameter κ which plays therole of a ‘coupling constant’. Without loss of generality, 0 � κ � 1 and we assume thatpn : [0, 1] −→ R+ is a continuous function for each j . Furthermore, as far as the ‘sparseness’condition is concerned, (pn)n�0 is required to be of the form

pn ={

1 − rj if n = aj ∈ A,

1 if n /∈ A,, (1.2)

with

rj (0) = 0 and rj (1) = 1, (1.3)

where A = {aj }j�1 is a set of strictly positive integers such that

aj − aj−1 � 2, j = 2, 3, . . . (1.4)

and

limj→∞

aj+1

aj

= β > 1 , (1.5)

where β is the ‘sparseness parameter’.A concrete example satisfying (1.3) is given by

rj = κ

1 + (1 − κ)ζj

(1.6)

with ζj � 0. If ζj → ∞ as j → ∞, then

rj (κ) → 0, (1.7)

for all κ ∈ [0, 1) and rj (1) = κ . If ζj = 0 for all j , then

rj (κ) = κ. (1.8)

In both cases (either ζj → ∞ or ζj = 0), as κ varies in the interval (0, 1), J interpolatescontinuously two opposite situations: J has a pure a.c. spectrum at κ = 0 and a purely pointspectrum at κ = 1. Note that, in both cases, pn(1) = 0 for all n ∈ A and J becomes a directsum of finite matrices with a dense p.p. spectrum if (1.5) holds. This is Howland’s remark [H],which shows off-diagonal Jacobi matrices as natural examples in which a spectral transitionmight be expected. Such transitions also occur, however, in the diagonal case (see theorem 6.3of [Z] for an example in the random case).

Spectral transition in a sparse model and a class of nonlinear dynamical systems 767

Jacobi matrices (1.1) can be written as

( Ju)n = pn un+1 + pn−1 un−1, (1.9)

n ∈ N, with p−1 = 1 fixed and boundary condition u−1 = 0. A Jacobi matrix Jα is said tosatisfy a α-boundary condition at −1 if u−1 is given by

u−1 cos α − u0 sin α = 0 (1.10)

(J = J0 satisfies a Dirichlet 0-boundary condition at −1, see also (2.12)). We shall call theclass of Jacobi matrices (1.9) satisfying (1.10) for some α ∈ [0, π) and (1.2 )–(1.5) Howland’sclass H and the subset of those satisfying (1.7) H0. Note that H\H0 include those in which(1.8) holds.

Let J 0 be the free Jacobi matrix

J 0 =

0 1 0 0 · · ·1 0 1 0 · · ·0 1 0 1 · · ·0 0 1 0 · · ·...

......

.... . .

.

The spectrum σ(J 0) = [−2, 2] is purely absolutely continuous. If J ∈ H0, Q = J − J 0 isa compact perturbation of J 0 and thus, by Weyl’s invariance theorem, the essential spectrumof J remains the same as that of J 0, σess(J ) = σ(J 0) = [−2, 2]. The spectrum of J in thissubclass is described as follows.

Theorem 1.1. Let J ∈ H0 with 0 < κ < 1. Then σ(J ) is purely a.c. if∑∞

k=1 |rk|2 < ∞ andpurely s.c. if

∑∞k=1 |rk|2 = ∞ and the ‘sparseness parameter’ is β = ∞.

Theorem 1.1 may be proven along the lines of theorem 1.7 of Kiselev–Last–Simon [KLS](see also [KR] and references therein). It shows that there is no spectral transition in thesubclass H0 as κ varies in (0, 1), at least under the sparseness condition (1.5). In order fora transition to occur at a nontrivial value of κ , J has to be such that (1.7) does not hold (i.e.rn � 0) and it is likely to be from s.c. to (dense) p.p., with a.c. spectrum just for κ = 0.

When rn � 0, i.e. (1.8), theorem 1.1 is no longer applicable. This is the case we considerin the rest of the paper.

The literature on sparse potentials is extensive, but one of the very basic methods employedto study the spectrum is given by Pearson [P]. The idea is that sparse ‘bumps’ ai lead to the‘independence’ of certain (deterministic) functions which behave as (functions of) a randomvariable, uniformly distributed (u.d.) over an interval of energy (see the discussion on [P],p 21). For a recent reference along these lines see [BP], in particular section 5.

In this paper, we introduce a different method, in which the emphasis is on u.d. on thespace-variable, rather than on the energy. More precisely, we prove in section 4 (theorem4.4) that if the Prufer angles (θk)k�1, where k is the space-variable, are uniformly distributedfor almost every value of the energy, there are two complementary sets of parameters (β, p),both of positive Lebesgue measure, in which the spectra are, respectively, s.c. and (dense) p.p.Similar results also hold if a weaker conjecture (conjecture 4.1) is valid (see remark 4.5.3).

In section 5 the perturbation scheme introduced in [Z] is extended to show that, withinour framework, one of the assumptions of theorem 4.4 may be verified if β is sufficientlylarge. This relies strongly on the metric theorems of ergodic theory [KN]. As a corollary,one obtains a s.c. spectrum on a set of positive Lebesgue measure (theorem 5.8). In section 6we provide compelling numerical evidence of the validity of conjecture 4.1, again supportedby the metric theorems. Section 2 presents very briefly some results on the spectrum, and

768 D H U Marchetti et al

section 3 introduces the Prufer variables. Section 7 is reserved for a conclusion, with a briefstatement on open problems.

2. Some notation and a theorem on the spectrum

A bounded Jacobi matrix J is a matrix of a self-adjoint operator H on a separable Hilbert spaceH : Jij = (ϕi, Hϕj ), where {ϕj }j�0 is an orthonormal base of H . The spectral measureµ = dρ associated with J is given by the 00-element of the resolvent matrix

(J − λI)−100 = (ϕ0, (H − λ)−1ϕ0) =

∫ ∞

−∞

1

x − λdρ(x). (2.1)

We introduce, for λ ∈ C and n ∈ N, a 2 × 2 transfer matrix

T (n, n − 1; λ) = λ

pn

−pn−1

pn

1 0

, (2.2)

associated with the l2(N)-solutions u of (J − λI)u = 0 and set

T (n; λ) := T (n, n − 1; λ) T (n − 1, n − 2; λ) · · · T (0, −1; λ), (2.3)

with p−1 = 1. If J ∈ H, in view of (1.2) and (1.4), there are three situations to be considered:for n /∈ {aj , aj + 1}j�1, T (n, n − 1; λ) is equal to the free transfer matrix

T0(λ) =(

λ −11 0

), (2.4)

for n = aj + 1, for some j � 1,

T (n, n − 1; λ) = T+(aj ; λ) :=(

λ −1 + rj

1 0

)(2.5)

and for n = aj , for some j � 1,

T (n, n − 1; λ) = T−(aj ; λ) := λ

1 − rj

−λ

1 − rj

1 0

. (2.6)

Let λ = 2 cos ϕ where ϕ ∈ (0, π) (see theorem 2.1 for a proof that this parametrization isexhaustive for the essential spectrum which will be our only concern) and

U =(

0 sin ϕ

1 − cos ϕ

)(2.7)

be the matrix U that makes T0(λ) similar to a pure rotation

R(ϕ) =(

cos ϕ sin ϕ

− sin ϕ cos ϕ

)= UT0(λ)U−1. (2.8)

It will be most convenient to introduce, for each aj ∈ A, the matrices P+−(aj ) given by

T U+−(aj ) = R(ϕ)P+−(aj )R(ϕ), (2.9)

where T U+−(aj ) = UT+(aj )T−(aj )U

−1. From (2.3) to (2.9) and, since a product of rotations isa rotation, we have

UT (n; λ)U−1 = R((n − ak)ϕ)P+−(ak)R((ak − ak−1)ϕ) · · · P+−(a1)R((a1)ϕ), (2.10)

which will be useful in section 3.According to (1.2) and (1.8), in the above equations

rj = 1 − paj, (2.11)

where (paj)j�1 ≡ 0 refers to the p.p. case and (paj

)j�1 ≡ 1 to the unperturbed (a.c.) case.

Spectral transition in a sparse model and a class of nonlinear dynamical systems 769

Theorem 2.1. If J ∈ H and 0 � rj � 1 for all j ∈ N, i.e. paj∈ [0, 1], then

σess(J ) = [−2, 2]

is the essential spectrum of J .

Proof. With J0 given by (1.9) with the 0-boundary condition at −1, we have

Jα = J0 + tan α δ0. (2.12)

Clearly, (δ0u)n = δn0 u0 is a rank-one perturbation of J0 and σess(Jα) = σess(J0), by Weyl’sinvariance principle. Thus, the choice J = J0 entails no loss of generality, as far as the essentialspectrum is concerned.

We prove that σ(J0) = [−2, 2]. Thus, there is no discrete spectrum and σ(J0) = σess(Jα)

for any α ∈ [0, π)4.Let J0 ∈ H with (pn)n�0 so that 0 � paj

� 1, ∀j . To prove σ(J0) ⊆ [−2, 2], letu = (un)n�0 ∈ l2(N). We have

(u, J0u) =∞∑

n=0

(un un+1)hn

(un

un+1

), (2.13)

where

hn =(

0 pn

pn 0

).

Inserting

hn = λ+n P +

n + λ−n P −

n

into (2.13), where P ±n projects onto the eigenvector associated with the eigenvalue λ±

n = ±pn,we have

−2 λ � (u, J0u)

(u, u)� 2λ,

where λ = supn λ+n = supn pn = 1. Hence

σ(J0) ⊆ [−2 λ, 2 λ] = [−2, 2]. (2.14)

Note that, because J0 is a symmetric off-diagonal matrix, the spectrum is symmetric withrespect to the origin. In order to prove the opposite inclusion

σ(J0) ⊇ [−2, 2], (2.15)

we use Weyl’s criterion (see theorem VII.12 of [S1]): if A is a bounded self-adjoint operatorin separable Hilbert space H , λ is a point of the spectrum σ(A) of A if and only if there existsa sequence (ψn)n�0 in H , with ‖ψn‖ = 1, such that

limn→∞ ‖(A − λ) ψn‖ = 0 . (2.16)

Let λ = 2 cos ϕ, ϕ ∈ [0, π ] and define ψn = (1/√

n)(eiϕ, ei2ϕ, . . . , einϕ, 0, . . .). Clearlyψn ∈ l2(N) and we claim that

‖(J0 − λ)ψn‖ � cln n√

n(2.17)

holds with c = c(β) independent of n for any J0 ∈ H. In order to prove (2.17), note that(J0 − λ)ψn consists of the action on ψn of a sum of local matrices, each bounded in norm

4 Although Jα may have some discrete spectrum, it will not concern us in what follows.

770 D H U Marchetti et al

by one. Two of them involve the extremes eiϕ and einϕ , and there are O(ln n) nondiagonalmatrices, each acting just on two successive components of ψn. The O(ln n) comes from thesequence (aj )j�1 satisfying the sparseness condition (1.5), aj ∼ βj , with at most r points aj

within [1, n], where r is such that βr � n or, equivalently,

r � ln n

ln β.

This proves (2.17) and therefore (2.15). �Henceforth λ ∈ [−2, 2] will be a point of σess(J ).

3. Prufer variables

For λ ∈ C and n ∈ N, let T (n, n − 1; λ) be the 2 × 2 transfer matrix given by (2.2) and

note that if un =(

un

un−1

)is a vector in C

2 composed of the components of a l2(N)-solution

u = (u0, u1, . . .) of (Jα − λ)u = 0, then

un+1 = T (n, n − 1; λ) un (3.1)

holds for n = 0, 1, . . ., with an initial condition u0 =(

u0

u−1

)=(

cos α

sin α

)satisfying (1.10) for

some α ∈ [0, π ].We use equation (2.10) together with (3.1) and (2.3) to define the Prufer variables

(Rk, θk)k�0. For this, we need an expression for P+,−(a) at each value a ∈ A = {ak}k�1.Inserting

R(−ϕ)UT+(a)U−1 =(

pa 0(1 − pa) cot ϕ 1

)and

UT−(a)U−1R(−ϕ) =(

1 0(1/pa − 1) cot ϕ 1/pa

)into (2.9), where T±(a), U and R are given by (2.5)–(2.8), yields

P+−(a) =(

pa 0qa cot ϕ 1/pa

)(3.2)

with qa = 1/pa − pa . If

vk = (Rk−1 cos θk, Rk−1 sin θk) (3.3)

and

vk =(Rk cos θk, Rk sin θk

)(3.4)

are defined by

vk = R ((ak − ak−1)ϕ) vk−1 (3.5)

and

vk = P+− (ak) vk , (3.6)

with a0 = 0 and the initial condition

v0 =(

cos θ0

sin θ0

)= U

(cos α

sin α

)=(

sin α sin ϕ

cos α − sin α cos ϕ

)(3.7)

Spectral transition in a sparse model and a class of nonlinear dynamical systems 771

then, we have

Uuak= vk = vk−1 .

Since ∂θ0/∂α = − sin ϕ, for each ϕ ∈ (0, π) there is a one-to-one correspondence betweenthe initial Prufer angle θ0 and the α-boundary condition at −1 of (1.9). It follows that theEuclidean norm of UT (n; λ) U−1, given by∥∥UT (n; λ) U−1v0

∥∥2 = ∥∥UT (aN ; λ) U−1v0

∥∥2 = R2N, (3.8)

for any n such that aN � n < aN+1, can thus be written as

R2N = R2

N

R2N−1

R2N−1

R2N−2

· · · R21

R20

=(

exp

{1

N

N∑k=1

(f(pak

, θk

) − ln p2ak

)})N

, (3.9)

where

f (p, θ) = ln(p4 cos2 θ +

(sin θ +

(1 − p2

)cot ϕ cos θ

)2)

(3.10)

with the Prufer angles (θk)k�0 satisfying the recursion relation

θk+1 = g(pak, θk) + (ak+1 − ak) ϕ, (3.11)

where

g(p, θ) = tan−1

(1

p2(tan θ + cot ϕ) − cot ϕ

), (3.12)

for k � 1, with θ1 given by

θ1 = θ0 + a1ϕ . (3.13)

4. The main theorem

In this section we prove the main theorem (theorem 4.4) connecting continuous asymptoticdistribution of the Prufer angles with the spectral transition. We combine proposition 4.3 statedbelow with the powerful theorems of Last and Simon [LS] relating the nature of the spectrumof J to the properties of the transfer matrix T (n; λ): the essential support �ac of the a.c. partµac of spectral measure µ = dρ (2.1) is given by

�ac ={

λ : lim infN→∞

1

N

N−1∑n=0

‖T (n; λ)‖2 < ∞}

. (4.1)

By theorems 1.6 and 1.7 of [LS] the eigenvalue equation (J − λI)u = 0 has no l2(N)-solutions if

∞∑n=0

‖T (n; λ)‖−2 = ∞. (4.2)

(see also theorem 2.1 in [SS]) and an l2(N)-solution if

Npp ≡∞∑

n=0

‖T (n; λ)‖2

( ∞∑k=n

‖T (k; λ)‖−2

)2

< ∞ . (4.3)

772 D H U Marchetti et al

We say that the Prufer angles (θk)k�1 defined by (3.3) have an asymptotic distributionfunction modulo π (abbreviated a.d.f. mod π ) ν(x) if, for any θ ∈ [0, π ],

limN→∞

card ({k : θk ∈ [0, θ), 1 � k � N})N

= ν(θ), (4.4)

where card(S), the cardinality of the set S, is the number of elements in S. Note that thefunction ν : [0, π ] → [0, 1] is nondecreasing right continuous with ν(0) = 0 and ν(π) = 1.The sequence (θk)k�1 need not have an a.d.f mod π but upper

lim supN→∞

card ({k : θk ∈ [0, θ), 1 � k � N})N

= �+(θ)

and lower

lim infN→∞

card ({k : θk ∈ [0, θ), 1 � k � N})N

= �−(θ)

distribution functions, with 0 � �−(θ) � �+(θ) � 1 for θ ∈ [0, π ], always exist. If�− ≡ �+, the sequence (θk)k�1 has an a.d.f. mod π . If �−(θ) = �+(θ) = θ for θ ∈ [0, π ],the sequence (θk)k�1 is uniformly distributed modulo π (abbreviated u.d. mod π ).

For simplicity, and without loss of generality, we fix here and henceforth pak= p for all

k, 0 < p < 1, and take A = (ak)k�1 so that

ak − ak−1 = βk (4.5)

with a0 = 0, i.e. ak = ak(β) = (βk+1 − β

)/(β − 1), β > 1. In this case (3.9) and (3.11) read

R2N =

(1

p2exp

{1

N

N∑k=1

f (θk)

})N

(4.6)

with

θk = g(θk−1) + βkϕ, (4.7)

where f (θ) = f (p, θ) is given by (3.10), g(θ) = g(p, θ) by (3.12) and θ1 by (3.13).

Conjecture 4.1. The sequence (θk)k�1 has the continuous a.d.f. mod π , ν(θ), for ϕ ∈[0, π ]\Aθ0 , where Aθ0 is a set of Lebesgue measure zero, possibly θ0-dependent.

We have (see theorem 7.2, chapter 1 of [KN]) the following theorem.

Theorem 4.2. A sequence ω = (θk)k�1 of real numbers has the continuous a.d.f. mod π , ν(θ),if and only if

limN→∞

1

N

N∑k=1

f (θk) = 1

π

∫ π

0f (θ) dν(θ) (4.8)

holds for every continuous periodic function f of period π .

The other ingredient is the following result.

Proposition 4.3. Suppose

[a, b] ⊂ σ (Jα) (4.9)

holds for all α ∈ [0, π ] and there exists A ⊂ [a, b] of Lebesgue measure zero independent ofα such that, if λ ∈ [a, b]\A,

(J − λI) u = 0 (4.10)

has a solution u ∈ l2(N), where J is given by (1.9) for n = 1, 2, . . ., and u0 may take anyvalue. Then σ(Jα) has only dense p.p. spectrum in [a, b] for a.e. (w.r.t. Lebesgue measure)α-boundary condition (1.10).

Spectral transition in a sparse model and a class of nonlinear dynamical systems 773

Proof. Take λ ∈ [a, b]\A. By (4.10), (1.9) has, for this λ, a l2-solution on (0, ∞). Thus, bytheorem II.3 of [S2], either (1) λ is an eigenvalue of J with Neumann b. c. (i.e. correspondingto u−1 = 0 in (1.10) or α = 0) or (2) G(λ) = ∫

dρ(y)/(λ − y)2 < ∞, where ρ is the spectralmeasure (2.1). The first alternative (1) can hold at most for λ belonging to a countable set,but, by hypothesis, (4.10) holds in a set of positive Lebesgue measure, i.e. the complementof a set of zero Lebesgue measure. Thus, even excluding the countable set satisfying the firstalternative, we are led to the statement that G(λ) < ∞ for a.e. λ in [a, b] and therefore, bythe Simon–Wolff criterion [SW] (see also theorem II.5 of [S2]), Jα has only p.p. spectrumin [a, b] for a.e. boundary condition α, which is a nontrivial statement by assumption (4.9).Finally, the inclusion of those λ satisfying the first alternative does not alter the statement ofthe proposition. �

The assumption of the following theorem—our main result—holds for certain limit valuesof parameters p, β and λ = 2 cos ϕ, but the scenery described differs only slightly in theirneighbourhood (see section 6 for compelling numerical evidence and theorem 5.8 for a rigorousresult).

Theorem 4.4. Let Jα ∈ H\H0 with rj = 1−p for all j ∈ N, p ∈ (0, 1) and β ∈ N, β � 2. Let

I1 ≡{

λ ∈ [−2, 2] :

(p

1 − p2

)2

(β − 1)(4 − λ2

)> 1

}(4.11)

and

I2 ≡{

λ ∈ [−2, 2] :

(p

1 − p2

)2

(β3 − 1)(4 − λ2) < 1

}. (4.12)

If the Prufer angles (θk)k�1 are uniformly distributed modulo π for ϕ ∈ [0, π ]\Aθ0 , where Aθ0

is a set of Lebesgue measure zero, possibly θ0-dependent, then

(a) there exists a set A1 of Lebesgue measure zero such that the spectrum restricted to the setI1\A1 is purely singular continuous,

(b) the spectrum σ(Jα) of Jα is a dense pure point when restricted to I2 for almost everyα ∈ [0, π), where α characterizes the boundary condition (1.10).

Remark 4.5.

1. Take, e.g., β = 2: (4.11) holds for p sufficiently close to 1, i.e. for small ‘coupling’κ = 1−p, and λ never in a neighbourhood U near the edges of the band ±2, while (4.12)holds for p sufficiently close to zero (‘large’ κ) and λ always close to the band edges (seefigure 1). These features are also expected in the Anderson model, with the continuouspart of the spectrum being, however, absolutely continuous.

2. In section 6 we present numerical evidence that the sequence (θk)k�1 has a continuousa.d.f. mod π for any parameter values p ∈ (0, 1) and β = 2 and becomes u.d. mod π inthe limit as p goes to both 0 and 1.

3. The proof of theorem 4.4 holds even when the Prufer angles are not nearly uniformlydistributed under the assumption that the corresponding lhs of the integral (4.17), withdν(θ) replacing dθ , is close to the rhs of (4.17). If (θk)k�1 has a continuous a.d.f. mod π ,the rhs of (4.8) with f given by (4.14) can be evaluated numerically in order to determineintervals I1 and I2 similarly to ( 4.11) and (4.12). Figures 2 and 3 show the profile of theintegral as a function of ϕ for p = 0.1 and p = 0.3. The dots represent the numericalintegration of (4.17) with the Lebesgue measure dθ replaced by the Stieltjes–Lebesguemeasure dν where ν is the corresponding distribution given in figure 4.

774 D H U Marchetti et al

Figure 1. Intervals supporting s.c. (light grey) and p.p. (medium grey) spectra of J for β = 2 andp = 1/2.

Figure 2. Profile of the integral (4.17) as a function of ϕ for p = 0.1.

4. Theorem 4.4 holds without the uniform distribution condition if A = (aj )j�1 in (1.2)is given by a sequence of independent random variables, defined in a probability space(�, ν), with aj = aj (ω) uniformly distributed in {βj − j, . . . , βj + j}. It follows fromsection 6 of [Z] that the lhs of (4.13) converges to 0 for almost every realization A(ω) andthe inequalities in I1 and I2 may be sharpened as in theorem 6.3 of [Z].

Proof. We must verify that the conditions on p, β and λ = 2 cos ϕ which define (4.11) and(4.12) imply (4.2) and (4.3), respectively, for some J = Jα satisfying the α-boundary conditionat −1.

By assumption (4.8) the large N -behaviour of RN is governed by (4.6) with the sum inthe exponent substituted by the integral (4.8). More precisely, Koksma’s inequality gives (seetheorem 5.1 and corollary 1.1 in chapter 2 of [KN])5∣∣∣∣∣ 1

N

N∑k=1

f (θk) − 1

π

∫ π

0f (θ) dθ

∣∣∣∣∣ � D∗N

π

∫ π

0

∣∣f ′(θ)∣∣ dθ, (4.13)

5 The Lebesgue measure dθ may be substituted by the Riemann–Stieltjes measure dν(θ) in this theorem.

Spectral transition in a sparse model and a class of nonlinear dynamical systems 775

Figure 3. Profile of the integral (4.17) as a function of ϕ for p = 0.3.

Figure 4. Histograms with 107 iteration points of the map (4.7) for β = 2, ϕ a dyadic irrationaland p2 = 0.01, 0.1, 0.3, 0.7, 0.9 and 0.99, respectively.

(This figure is in colour only in the electronic version)

where the discrepancy

D∗N := sup

0<α�π

∣∣∣∣card({k : θk ∈ [0, α), 1 � k � N})N

− α

∣∣∣∣goes to 0 as N goes to ∞.

776 D H U Marchetti et al

In order to employ theorem 4.2 it is necessary to ascertain that f is a Riemann-integrableperiodic function of period π and compute its integral. By simple trigonometric identities(3.10) may be written in the form

f (θ) = ln(a + b cos 2θ + c sin 2θ), (4.14)

where

2a = (1 − p2)2 cot2 ϕ + 1 + p4,

2b = (1 − p2)2 cot2 ϕ − 1 + p4

and

c = (1 − p2) cot ϕ,

which shows that f is periodic with period π . An explicit computation gives

a + b cos 2θ + c sin 2θ � minθ

(a + b cos 2θ + c sin 2θ)

= a −√

b2 + c2 = a −√

a2 − p4 > 0, (4.15)

since

0 < b2 + c2 = a2 − p4 (4.16)

if 0 < p < 1, and this implies that f (θ) is Riemann-integrable. From [PBM, p 546], (4.14)and (4.16), we have

1

π

∫ π

0f (θ) dθ = ln

(a +

√a2 − b2 − c2

2

)= ln

((1 − p2)2cosec2ϕ

4+ p2

). (4.17)

Now let,

r (p, ϕ) ≡ 1 +(1 − p2)2cosec2ϕ

4p2. (4.18)

By (4.6), (4.13), (4.14) and (4.17),

C−1N rN � R2

N � CN rN, for ϕ ∈ [0, π ]\Aθ0 (4.19)

where Aθ0 is a set of Lebesgue measure 0, possibly depending on θ0, mentioned in thehypothesis, and some finite constant CN > 1 such that limN→∞ C

1/N

N = 1. Note thatsupθ∈[0,π ]

∣∣f ′(θ)∣∣ < ∞ in view of (4.15), so that the rhs of (4.13) indeed yields a CN with the

stated properties.Let ‖A‖U := ∥∥UAU−1

∥∥ be a matrix norm defined by the invertible matrix U given by(2.7) with ϕ ∈ (0, π), where

‖B‖ = supv∈C2

‖Bv‖‖v‖ .

Let n be such that aN � n < aN+1 holds for some N ∈ N. We have∑m�n

‖T (m; λ)‖−2U =

∑k�N+1

‖T (ak; λ)‖−2U βk+1 + ‖T (aN ; λ)‖−2

U

∑n�m<aN+1

1. (4.20)

By the method in [KLS, lemma 2.2 and theorem 2.3], it can be proved that

‖T (m; λ)‖−2 � C2 ‖T (ak; λ)‖−2U � C2

(maxi∈{1,2}

Rk(θi0)

)−2

(4.21a)

Spectral transition in a sparse model and a class of nonlinear dynamical systems 777

and

‖T (m; λ)‖−2 � C−2 ‖T (ak; λ)‖−2U � C−2

(maxi∈{1,2}

Rk(θi0)

)−2

(4.22a)

hold with C = √(1 + |cos ϕ|) / (1 − |cos ϕ|) and C = C/

∣∣sin(θ1

0 − θ20

)/2∣∣ for any pair

θ10 , θ2

0 ∈ [0, π ] with 0 <∣∣θ1

0 − θ20

∣∣ < π/2. Now let,

S±N,M ≡ β

M∑k=N

C±1k

(β

r

)k

. (4.23)

By (4.19)–(4.23),

C−2 S−N+1,M �

aM+1∑m=n

‖T (m; λ)‖−2 � C2 S+N,M (4.24)

holds for all integers M � N + 1 and λ ∈ B where6

B ≡ 2 cos([0, π ] \A′) (4.25a)

and

A′ = Aθ10∪ Aθ2

0(4.26a)

is a fixed set of Lebesgue measure zero, once we have chosen an arbitrary but from now onfixed set

{θ1

0 , θ20

}. The further assertions of the lemma depend solely on the bounds (4.24).

By (4.18), (4.19) and (4.25a)

B ⊆{λ : lim

n→∞ ‖T (n; λ)‖ = ∞}

(4.27)

and lies in the complement of �ac by (4.1). Thus, except for a set

A1 = A′ ∪ A′′ (4.28)

of Lebesgue measure zero—where A′′ is a set of zero Lebesgue measure connected with thedefinition of the essential support �ac of µac (see, e.g., [CFKS, p 179], for this definition)—the set

B1 ≡ 2 cos ([0, π ] \A1) (4.29)

lies in the singular spectrum. By (4.23) and the lhs of (4.24), (4.2) holds (taking M → ∞) if

4p2(β − 1) > (1 − p2)2cosec2ϕ , (4.30)

which, using 4cosec−2ϕ = 4 − λ2, may be written in the form (4.11) and defines I1. Thus, by(4.29) the spectrum is purely singular continuous when restricted to the set

I1\ (A1 ∩ I1) ,

which is assertion (a) of theorem 4.4.We now prove assertion (b), assuming that

4p2(β − 1) < (1 − p2)2cosec2ϕ, (4.31)

which, by (4.18) and (4.23), implies that S+N,M converges for each N as M → ∞. Using again

(4.24) and denoting by

S+N = lim

M→∞S+

N,M, (4.32)

6 Here, f (A) denotes the set {f (x) : x ∈ A}.

778 D H U Marchetti et al

we find∞∑

m=n

‖T (m; λ)‖−2 � C2 S+N . (4.33)

By (4.19), (4.21a), (4.32) and (4.33), and recalling the notation in (4.3), there exist finiteconstants C ′ and C ′′ such that

Npp � C ′∞∑

N=1

βNR2N

(S+

N

)2 � C ′′∞∑

N=1

C3N

(4p2β3

(1 − p2)2cosec2ϕ + 4p2

)N

.

The above series converges if

p2

(1 − p2)2(β3 − 1)(4 − λ2) < 1, (4.34)

which defines I2 in (4.12). Thus, by (4.3) and (4.29) if λ belongs to the set

B2 = I2\ (A1 ∩ I2)

the eigenvalue equation (J − λI) u = 0 has an l2 (N)-solution. Since

I2 ⊂ σ (Jα) , ∀α ∈ [0, π),

by theorem 2.1, the assumptions of proposition 4.3 hold and Jα has only (dense) p.p. spectrumin I2 for almost every α ∈ [0, π), concluding the proof of assertion (b) of theorem 4.4. �

Remark 4.6. It follows from a general result of rank-one perturbations (see, e.g., theoremII.11 in [S2]) that, for a dense Gδ set of α-boundary conditions at −1, Jα has a purely s.c.spectrum on I2. The Gδ set in [0, π ] has, by proposition 4.3, Lebesgue measure zero.

5. Uniform distribution of Prufer angles

In this section the conclusions of theorem 4.4 will be extended beyond its assumption onuniform distribution of Prufer angles by applying the perturbation scheme of [Z] to the problemat hand. The main result is theorem 5.8. We observe that the proof of theorem 4.4 remainsunaltered if the Prufer angles {θk} are replaced by an uniformly distributed sequence {ξk}. By(4.6), (4.13) and (4.19), we need only evaluate the error

EN = 1

N

N∑k=1

(f (θk) − f (ξk)) (5.1)

of replacing Birkhoff averages associated with {θk} by one with {ξk} and show that it can bemade so small that, with a slight modification in the spectral conditions (4.11) and (4.12), theconclusions of theorem 4.4 hold.

Given a sequence of numbers (ξk)k�1 let (Sn)n�1 be given by

Sn = 1

n

n∑k=1

e2ihξk . (5.2)

According to the Weyl criterion [KN, theorem 2.1 of chapter 1], the sequence (ξk)k�1 is u.d.mod π if and only if lim

n→∞ Sn = 0 for every integer h �= 0. If ξk = ξk(x) is defined for each

x lying in some interval [a, b] ⊂ R, then Sn = Sn (x) is also defined for each x ∈ [a, b].

Denoting by ‖f ‖L2([a,b]) =(∫ b

a|f (x)|2 dx

)1/2the L2 norm, the general metric result is as

follows (see theorem 4.2, chapter 1, of [KN] for a proof).

Spectral transition in a sparse model and a class of nonlinear dynamical systems 779

Theorem 5.1. If the numerical series∞∑

n=1

1

n‖Sn‖2

L2([a,b])

converges for each integer h �= 0 then the sequence {ξk} is u.d. mod π for almost all x ∈ [a, b],i.e. {ξk} is u.d. mod π for every x lying in [a, b] apart from a set with Lebesgue measure 0.

Now, let {ξn}n�1 be a sequence of continuous piecewise linear functions ξn = ξn(ϕ) ofϕ ∈ [0, π ] which coincide with θn = θn(ϕ) on every value ϕn,j at which θn(ϕn,j ) is a multipleof π . Zlatos (see appendix A of [Z]) has introduced this sequence to estimate the Hausdorffdimension of invariant subsets of [0, π ] for which the Birkhoff averages do not exist. We showhere that the general metric theorem can be applied to it. For this, we need some propertiesof (3.12).

Proposition 5.2. The function g given by (3.12) maps [−π/2, π/2] onto itself and has threefixed point solutions: θs

± = ±π2 are two stable solutions and θu = ϕ − π

2 , ϕ ∈ (0, π), isunstable. g is monotone increasing with g′(θs

±) = p2, g′(θu) = 1/p2 and there are twoinflection points θ i

± such that

g′(θ i−) = c1p

2 � g′(θ) � c2

p2= g′(θ i

+) (5.3)

holds for ci = ci(p, ϕ) = 1 + O(p2), i = 1, 2.

Proof. Writing g(θ) = tan−1(ζ/p2

)where

ζ(θ) = tan θ + (1 − p2) cot ϕ, (5.4)

it follows from

g′(θ) = p2

p4 + ζ 2sec2 θ (5.5)

that g is a monotone increasing function with g(−π/2) = −π/2 and g(π/2) = π/2. Henceg : [−π/2, π/2] → [−π/2, π/2]. As θ goes to ±π/2, sec2 θ/ζ 2 → 1 and g′(θ) → p2.Since p < 1, θ = ±π/2 are stable fixed points of g. To find the unstable fixed point weobserve that

tan g(θ) + cot ϕ = 1

p2(tan θ + cot ϕ) ,

together with the fixed point equation g(θ) = θ and p < 1, implies

tan θu + cot ϕ = 0, (5.6)

which is equivalent to θu = ϕ − π/2. At θ = θu equation (5.4) reads ζ(θu) = p2 cot ϕ whichtogether with sec2 θu = 1 + tan2 θu = 1 + cot2 ϕ and (5.5) gives

g′(θu) = 1

p2.

A direct computation yields

g′′(θ) = 2p2 sec2 θ(p4 + ζ 2

)2

[(p4 + ζ 2) tan θ − ζ sec2 θ

], (5.7)

from where we get the two inflection points by solving (p4 + ζ 2) tan θ − ζ sec2 θ = 0 for tan θ :

θ i± = tan−1

(∓ (cot ϕ)±1

(1 ± p2

sec2 ϕ+ O

(p4)))

(5.8)

780 D H U Marchetti et al

gives θ i+ = θu + O(p2) and θ i

− = ϕ + O(p2), if ϕ ∈ [0, π/2), and θ i− = ϕ − π + O(p2) if

ϕ ∈ [π/2, π). Equation (5.3) follows by perturbation around p = 0 and this concludes theproof of proposition 5.2. �

If p2β > c2, then (4.7) and (5.3) imply

θ ′k(ϕ) = g′(θk−1)θ

′k−1(ϕ) + βk

� c2

p2θ ′k−1 + βk � 1

1 − c2/(p2β

)βk , (5.9)

by induction. Analogously,

θ ′k(ϕ) � c1p

2θ ′k−1(ϕ) + βk

� 1

1 − c1p2/ββk. (5.10)

Hence, θk = θk(ϕ) is strictly monotone increasing function of ϕ ∈ [0, π ] satisfying θ ′k(ϕ)/βk ∈

[d1, d2] with d1 = d1(p, β, ϕ) = c1p2/β > 0 and d2 = d2(p, β, ϕ) = c2/

(p2β

)< ∞.

For each k ∈ N and interval I = (a, b] ⊂ [0, π ], let

ϕk,0 � a < ϕk,1 < ϕk,2 < · · · < ϕk,nk� b < ϕk,nk+1,

nk = nk(I ), be such that

θk

(ϕk,i

) =([

θk(a)

π

]+ i

)π (5.11)

holds for i = 0, 1, . . . , nk + 1 with [x] the integer part of x ∈ R. We set Ik,0 = (a, ϕk,1],Ik,i = (ϕk,i , ϕk,i+1] for i = 1, . . . , nk − 1, Ik,nk

= (ϕk,i , b] and note that{Ik,i

}nk

i=0 is a partitionof I with ∣∣Ik,i

∣∣ ∈[

π

d2βk,

π

d1βk

](5.12)

for every i �= 0, nk . Let {ξk}k�1 be a sequence defined by the linear interpolation of θk at thepoints

{ϕk,i

}:

ξk(ϕ) = θk

(ϕk,i

)+

ϕ − ϕk,i∣∣Ik,i

∣∣ π, ϕ ∈ Ik,i . (5.13)

Theorem 5.3. {ξk(ϕ)}k�1 is u.d. mod π for almost every ϕ ∈ [0, π ], any p ∈ (0, 1) andβ � β0(p, ϕ) � 2 large enough.

Proof. To prove theorem 5.3, it is enough to verify the assumption of theorem 5.1. For h ∈ Z

and I = (a, b] ⊂ [0, π ], we have

‖Sn‖2L2([a,b]) = 1

n2

n∑k,m=1

∫ b

a

exp {2ih (ξm(ϕ) − ξk(ϕ))} dx

� |b − a|n

+2

n2

∑1�k<m�n

∣∣∣∣∫ b

a

exp {2ih (ξm(ϕ) − ξk(ϕ))} dϕ

∣∣∣∣ . (5.14)

Note that, by (5.12),∣∣Im,j

∣∣ <∣∣Ik,l

∣∣ holds uniformly for k < m if β is large enough and thereis at least one l = l(j) such that Ik,l ∩ Im,j �= ∅. Using∫

Im,j

exp {2ihξm(ϕ)} dϕ =∫

Im,j

exp

{2ih

ϕ − ϕm,j∣∣Im,j

∣∣ π

}dϕ

= ∣∣Im,j

∣∣ ∫ 1

0e2ihxπ dx = 0

Spectral transition in a sparse model and a class of nonlinear dynamical systems 781

together with∣∣eiy − 1

∣∣ � |y|, for any y ∈ R and (5.12),∣∣∣∣∣∫

Im,j

exp {2ih (ξm(ϕ) − ξk(ϕ))} dϕ

∣∣∣∣∣ =∣∣∣∣∣∫

Im,j

(exp

{−2ih

(ξk(ϕ) +

ϕk,l∣∣Ik,l

∣∣π)}

− 1

)

× exp

{2ih

(ξm(ϕ) +

ϕk,l∣∣Ik,l

∣∣π)}

dϕ

∣∣∣∣∣� 2 |h| π∣∣Ik,l

∣∣∫

Im,j

ϕ dϕ = |h| π∣∣Im,j

∣∣2∣∣Ik,l

∣∣ (5.15)

and so∣∣∣∣∫ b

a

exp {2ih (ξm(ϕ) − ξk(ϕ))} dϕ

∣∣∣∣ �nk∑

j=0

∣∣∣∣∣∫

Im,j

exp {2ih (ξm(ϕ) − ξk(ϕ))} dϕ

∣∣∣∣∣� |h| π sup

i

∣∣Im,i

∣∣∣∣Ik,l(i)

∣∣ nk∑j=0

∣∣Im,j

∣∣� |h| πd2

d1βm−k |b − a| . (5.16)

Substituting (5.16) into (5.14) yields

‖Sn‖2L2([a,b]) �

(1 +

2 |h| πd2

d1(β − 1)

) |b − a|n

,

which proves the assumption of theorem 5.1 and concludes the proof of the theorem. �

Remark 5.4.∣∣Im,j

∣∣ <∣∣Ik,l

∣∣ holds uniformly for k < m if p4β > c1/c1 (� 1 for p small, seeproposition 5.2). The condition has been used in the proof for the integral (5.15) over Im,j tobe uniformly bounded, including the intervals with j = 0, nm at the extremities, which couldbe handled separately. Theorem 5.3 holds without this condition.

To estimate the (pointwise) distance between ξk(ϕ) and θk(ϕ), let{ϕk,j

}nk

j=0 be the set ofall values satisfying ξk(ϕk,j ) = θk(ϕk,j ). Note that this includes points that were not in theprevious list, since θk(ϕk,j ) may not necessarily be a multiple of π . From (4.7), (5.13) and∣∣Ik,j

∣∣ = ϕk,j+1 − ϕk,j , this difference can be written as

θk(ϕ) − ξk(ϕ) = βkϕ + g(θk−1(ϕ), ϕ) −(

ϕ − ϕk,j∣∣Ik,j

∣∣ θk

(ϕk,j+1

)+

ϕk,j+1 − ϕ∣∣Ik,j

∣∣ θk

(ϕk,j

))

= gk−1(ϕ) − ϕ − ϕk,j∣∣Ik,j

∣∣ gk−1(ϕk,j+1) − ϕk,j+1 − ϕ∣∣Ik,j

∣∣ gk−1(ϕk,j ) (5.17)

for ϕ ∈ Ik,j . Here we make explicit the dependence of g on ϕ and use the shorthandgn(ϕ) ≡ g(θn(ϕ), ϕ).

Proposition 5.5.

|θk(ϕ) − ξk(ϕ)| �∣∣Ik,j

∣∣4

(maxϕ∈Ik,j

g′k−1(ϕ) − min

ϕ∈Ik,j

g′k−1(ϕ)

)(5.18)

holds for ϕk,j < ϕ � ϕk,j+1 and j = 0, . . . , nk .

782 D H U Marchetti et al

This is a refined (local) version of an upper bound given in appendix A of [Z]. We proveit in the case where g′′

k−1(ϕ) does not vanish in Ik,j . The general case follows in a similar way.

Proof. To maximize the difference (5.17) in absolute value, gk−1(ϕ) is replaced by a piecewiselinear continuous function:

hk−1(ϕ) :={h−

k−1(ϕ) = gk−1(ϕk,j ) + g′k−1(ϕk,j )

(ϕ − ϕk,j

)if ϕ ∈ (ϕk,j , ϕ],

h+k−1(ϕ) = gk−1(ϕk,j+1) − g′

k−1(ϕk,j+1)(ϕk,j+1 − ϕ

)if ϕ ∈ (ϕ, ϕk,j+1],

with ϕ defined by continuity:

h−k−1(ϕ) = h+

k−1(ϕ),

which, after some manipulations, gives

|θk(ϕ) − ξk(ϕ)| � x (1 − x)∣∣Ik,j

∣∣ ∣∣g′k−1(ϕk,j+1) − g′

k−1(ϕk,j )∣∣

with x = (ϕ − ϕk,j

)/∣∣Ik,j

∣∣ and, together with x (1 − x) � 1/4, proves the proposition. �

Theorem 5.6. Let J ∈ H\H0 as stated in theorem 4.4. Conclusions (a) and (b) of theorem 4.4,with intervals I1 and I2 replaced by

I ε1 ≡

{λ ∈ [−2, 2] :

(p

1 − p2

)2

(β − eε)(4 − λ2

)> eε

}(5.19)

and

I ε2 ≡

{λ ∈ [−2, 2] :

(p

1 − p2

)2

(β3 − e−3ε)(4 − λ2) < e−3ε

},

hold if the limit superior of (5.1) is bounded

E := lim supN→∞

EN � ε (5.20)

for some ε > 0. In addition, for each ε > 0 and (p, ϕ) ∈ (0, 1) × (0, π) there existβ0 = β0(ε, p, ϕ) sufficiently large such that (5.20) holds for β � β0.

Proof. By (3.10), (5.12), (5.13) and proposition 5.5

1

N

N0∑k=1

|f (θk) − f (ξk)| � maxθ∈[0,π ]

∣∣f ′(θ)∣∣ 1

N

N0∑k=1

|θk − ξk|

� maxθ∈[0,π ]

∣∣f ′(θ)∣∣ d2 − d1

4d1

N0

N

converges to 0 as N → ∞ for any N0 and it does not contribute to E. As a consequence,the contribution of partial derivative ∂g/∂ϕ to g′

k(ϕ) may, without loss, be disregarded fromthe estimate (5.18) for k � N0 and the error absorbed into the constant CN of (4.19). This,together with the fact that θk (ϕ) � g(βk−1ϕ, ϕ) + βkϕ for β and k large enough, leads to thefollowing scaling version of (5.18).

Let{θ i}n+1

i=0 be a sequence of numbers satisfying θ0 � −π/2 < θ1 < · · · < θn � π/2 <

θn+1 and such that the intervals I 0 = [−π/2, θ1], I i = (θ i, θ i+1], I n = (θn, π/2] form a

partition of [−π/2, π/2] satisfying

1 − δ1 � β∣∣I i

∣∣ � 1 + δ2 (5.21)

for some small fixed constants δ1 and δ2. If y : [−π/2, π/2] × [0, π ] −→ R is the continuouspiecewise linear function defined by

y(θ i, ϕ) = g(θ i, ϕ), i = 0, . . . , n + 1,

Spectral transition in a sparse model and a class of nonlinear dynamical systems 783

we have, analogously to the estimate (5.18),

|y(θ, ϕ) − g(θ, ϕ)| �∣∣I i

∣∣4

(maxθ∈I i

∂g

∂θ(θ, ϕ) − min

θ∈I i

∂g

∂θ(θ, ϕ)

)(5.22)

for θ ∈ I i which, in view of (5.21) and (5.5), is bounded from above by C/β for any(p, ϕ) ∈ (0, 1) × (0, π) provided β is sufficiently large.

By the fundamental theorem of calculus together with the scaling version (5.22) of (5.18),for every k > N0

|f (θk) − f (ξk)| � supmin(θk,ξk)<θ<max(θk,ξk)

∣∣f ′(θ)∣∣ |θk − ξk|

� sup0<θ1<π

∣∣f ′(θ1)∣∣ max

imaxθ∈I i

|y(θ, ϕ) − g(θ, ϕ)| < ε (5.23)

holds for every ε > 0 provided β > β0 with β0 = β0(p, ϕ, ε) sufficiently large.The spectral conditions defining I ε

1 and I ε2 are obtained as in the proof of theorem 4.4

with the inequalities β/r > 1 and βr (β/r)2 < 1 substituted by βe−ε/r > 1 andβreε (βeε/r)2 < 1, respectively. This concludes the proof of theorem 5.6. �

Remark 5.7. For each ε > 0 and (p, ϕ) ∈ (0, 1) × (0, π) there exists β1 = β1 (ε, p, ϕ)

sufficiently large such that I ε1 given by (5.19) is not an empty set for any β � β1.

We are now able to state the main result of this section. Theorem 5.6 together withremark 5.7 implies the following remark.

Theorem 5.8. Let J ∈ H\H0 as stated in theorem 4.4 and let β = β (ε, p, ϕ) = max (β0, β1)

for each ε > 0 and (p, ϕ) ∈ (0, 1) × (0, π), with β0, β1 as in theorem 5.6 and remark 5.7.Then, for every β � β there exists a set Aε

1 of Lebesgue measure zero such that the spectrumσ(J ) restricted to set I ε

1 \Aε1 is purely s.c.

Concerning I ε2 , we have, however, the following remark.

Remark 5.9.

1. f given by (3.10) has first derivative extremized by

f ′(θ−) � f ′(θ) � f ′(θ+) = −f ′(θ−) = 2

p2

|b |b| − c |c||√a2 + b2

, (5.24)

where a, b and c are given in (4.14). When ϕ approaches the boundary of (0, π) the upperbound (5.24) tends to infinity, and this turns I ε

2 into the empty set! More precisely, (5.23)and (5.24) imply that(

p

1 − p2

)2β

cot2 ϕ>

C

ε

holds for some C > 0 if(4 − λ2

)/λ2 = cot−2 ϕ is sufficiently small and this goes in the

opposite direction of the inequality in I ε2 .

2. For β a fixed large integer and p very small, the scaling description of (5.18) requirespartitions which do not satisfy (5.21). According to proposition 5.2, there is a smallinterval J containing the unstable fixed point θu = ϕ − π/2 such that g changes abruptlyfrom −π/2 + O

(p2)

to π/2 − O(p2)

and every partition interval there has to be oforder p2. Estimating (5.22) by Cp2, together with (5.23) and (5.24), does not yield (5.20)for any ε > 0 by choosing p small enough without evoking statistical properties of thesequence {θk}. In other words, the sign changes of (5.1) have to be taken into account.

Thus, by remark 5.9, the p.p. spectrum does not seem to be tractable by perturbativemethods. We turn to this problem in the next section.

784 D H U Marchetti et al

6. Numerical evidence for asymptotic distribution function

In this section we present the compelling numerical evidence that the Prufer angles have ana.d.f. mod π for almost every ϕ ∈ [0, π). We have computed the asymptotic density functionρ = dν/dθ for β = 2, ϕ a typically dyadic irrational and p2 varying over the values 0.01, 0.1and 0.3. The corresponding histograms presented in figure 4, each containing 107 iterationpoints of the map (4.7), indicate that the sequence (θk)k�1 has continuous a.d.f. mod π , ν(θ)

which tends to the identity θ when p → 0. These claims will now be examined numericallyusing two criteria whose background is given in the following.

The powerful metric result is ( [KN], theorem 4.3 of chapter 1) the followign theorem:

Theorem 6.1 (Koksma’s metric theorem). Let (ξk(x))k�1 be a sequence of real valuedcontinuously differentiable functions defined in an interval [a, b]. If for any two positiveintegers k �= m, ξ ′

k − ξ ′m is a monotone function of x such that

∣∣ξ ′k − ξ ′

m

∣∣ � K > 0 holdsuniformly in k , m and x ∈ [a, b], then (ξk)k�1 is u.d. mod π for almost all x ∈ [a, b].

Since ξ ′′k (ϕ) = 0 for almost every ϕ ∈ [0, π ], theorem 6.1 applies to (ξk)k�1 given by

(5.13) but cannot be applied for the sequence (θk)k�1 with p ∈ (0, 1). To see this and makeour numerical results understandable the proof of Koksma’s theorem is included.

Proof. By the second mean value theorem, there exists c such that a < c < b and∣∣∣∣∫ b

a

e2i(ξk(x)−ξm(x)) dx

∣∣∣∣ =∣∣∣∣ 1

2i

∫ b

a

de2i(ξk(x)−ξm(x))

ξ ′k(x) − ξ ′

m(x)

∣∣∣∣=

∣∣∣∣ 1

2i

(1

ξ ′k(a) − ξ ′

m(a)

∫ c

a

de2i(ξk(x)−ξm(x))

+1

ξ ′k(b) − ξ ′

m(b)

∫ b

c

de2i(ξk(x)−ξm(x))

)∣∣∣∣� 1∣∣ξ ′

k(a) − ξ ′m(a)

∣∣ +1∣∣ξ ′

k(b) − ξ ′m(b)

∣∣ . (6.1)

Ordering{ξ ′k(x)

}n

k=1 such that ξ ′1(x) < ξ ′

2(x) < · · · < ξ ′n(x) holds for all x, the difference in

any two consecutive numbers satisfies ξ ′k(x) − ξ ′

k−1(x) > K and we havek−1∑m=1

1∣∣ξ ′k(x) − ξ ′

m(x)∣∣ < 2

n∑m=1

1

mK<

2

Kln 3n .

This together with (5.14) yields

‖Sn‖2L2([a,b]) <

|b − a|n

+4

nKln 3n, (6.2)

which combined with theorem 5.1 concludes the proof. �In order to apply theorem 6.1, θ ′

k(ϕ) − θ ′m(ϕ) has to be a monotone function for each

interval [a, b] independent of k and m. But this is not true for the sequence (4.7) since thenumber of monotone intervals of θ ′

k − θ ′m in I ⊂ (0, π), with k < m fixed, for which we may

apply Koksma’s theorem, is proportional to βk and this number is compensated by the constantK of theorem 6.1 which, in view of equations (5.9) and (5.3), has the same k dependence:

θ ′k(ϕ) − θ ′

k−1(ϕ) = (g′(θk−1) − 1

)θ ′k−1(ϕ) + βk

� − (1 − p2

)θ ′k−1 + βk

� βp2 − c2 − p2 + p4

p2β − c2βk .

Spectral transition in a sparse model and a class of nonlinear dynamical systems 785

Figure 5. Plots of ‖Sn‖2L2(I ) for β = 2 and p = 0.3

According to the results shown in figure 4, Sn(ϕ) given by (5.2) with ξk replaced by θk(ϕ)

cannot converge to 0 for any ϕ ∈ [0, π ] if p ∈ (0, 1). Note that, by the Weyl criterion togetherwith (4.8),

limn→∞ Sn = lim

n→∞1

n

n∑k=1

e2ihθk(ϕ)

=∫

e2ihθdν(θ) ≡ ωh (6.3)

converges to 0 for every h �= 0 if and only if (θk)k�1 is uniformly distributed, i.e. ν(θ) = θ .Despite this, we have numerically tested the criterion of theorem 5.1 for the sequence

(θk)k�1. Figure 5 plots ‖Sn‖2L2(I ) for β = 2 and p = 0.3 as n varies from 103 to 104. We

have used the Monte Carlo method of integration for an interval I of length 2−990 aroundϕ = 0.893 75 with the standard deviation from the predicted value kept below 2%. Accordingto the best interpolation fit ‖Sn‖2

L2(I ) approaches a constant as fast as c1−c2n−0.9. For sequences

satisfying the assumptions of theorem 6.1 (see equation (6.2)), ‖Sn‖2L2(I ) converges to 0 as fast

as cn−1 ln n. For the additive part of the sequence (4.7), xk(ϕ) = βkϕ, or for the sequenceξk(ϕ) , the situation is even better with ‖Sn‖2

L2(I ) decaying as 1/n without the log correction.An extension of Koksma’s metric theorem to a nonuniform distribution ν is an open

problem. However, if somehow one is able to subtract its limit point ωh from Sn, the integralon the second line of (6.2) would, instead of behaving like (6.2), decay as fast as c2n

−0.9

according to the behaviour of figure 5 for c1 −‖Sn‖2L2(I ). Hence, the decay may be interpreted

as an indication that the double sum in (5.14) would grow like n1+δ , δ � 0.1, and the sequence(θk)k�1 would have continuous a.d.f. mod π for almost every ϕ ∈ [0, π ].

Finally, we verify the following criterion due to Wiener–Schoenberg (see theorem 7.5,chapter 1 of [KN]): (θk)k�1 has a continuous a.d.f. mod π if and only if ωh = lim

n→∞ Sn exists

for every positive integer h and

limH→∞

1

H

H∑h=1

|ωh|2 = 0.

Figure 6 is a log–log plot of∑H

h=1 |ωh|2 /H , with each point ωh obtained by 107 iteration of(4.7) and ϕ a fixed dyadic irrational, as H varies from 2 to 6000.

786 D H U Marchetti et al

Figure 6. Wiener–Schoenberg criterion.

7. Conclusion: open problems

In this paper we have introduced a new method to study the spectrum of sparse Jacobimatrices, based on u.d. (of the Prufer angles (θk)k�1) on the space-variable rather than on theenergy. Together with metric theorems, we were able to prove the existence of a s.c. spectrum(theorem 5.8). The proof of a (dense) p.p. spectrum seems, however, not to be accessibleto a perturbation treatment (remark 5.9), but under an assumption (conjecture 4.1) on thedistribution of (θk)k�1, theorem 4.4 yields the existence of dense p.p. for a complementary setof parameters and thus of a spectral transition.

In section 6 strong numerical evidence for conjecture 4.1—again supported by the metrictheorems—was provided. Its verification remains, however, an open problem in the theory ofnonlinear dynamical systems (n.d.s.). In fact, the Prufer angles are associated with a n.d.s.(3.11), (3.12) and (3.13) in the space-variable. This n.d.s. belongs to the class of perturbationsof the β-adic shift:

xk = β xk−1 mod1, β � 2 integer,

when the ak in (3.11) satisfy (4.5) (see also 4.7). Such perturbations have been considered inorder to investigate whether arithmetical numbers are normal to the base β (see [BC, L] andreferences therein) and are thus of independent interest. We hope that the present paper maystimulate further progress in the ergodic properties of these n.d.s.

Acknowledgments

We thank Professors B Simon, A Klein, A Lopes and the anonymous referee for instructivecomments. WW should like to thank Professors J S Howland and A Joye for fruitfulconversations and remarks. DHUM is partially supported by CNpq and FAPESP, WW byCNPq and LFG by FAPESP under Grant 03/13772-0.

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