Most linear flows on $\mathbb{R}^d$ are Benford

0.31 MB·English

by  Arno Berger

Most linear flows on Rd are Benford
5
1
0
Arno Berger
2

n Mathematical and Statistical Sciences
a
J University of Alberta

1 Edmonton, Alberta, Canada
2

] January 23, 2015
S
D
.
h Abstract
t
a
A necessary and sufficient condition (“exponential nonresonance”) is estab-
m
lished for everysignal obtained from alinear flow on Rd bymeansof alinear
[
observable to either vanish identically or else exhibit a strong form of Ben-

1 ford’s Law (logarithmic distribution of significant digits). The result extends
v
and unifies all previously known (sufficient) conditions. Exponential nonres-
5
4 onance is shown to be typical for linear flows, both from a topological and a
3 measure-theoretical point of view.
5
0 Keywords. Benford function, (continuous) uniform distribution mod 1, linear
.
1 flow, observable, Q-independence,(exponentially) nonresonant set.
0 MSC2010. 34A30, 37A05, 37A45, 11J71, 11K41, 62E20.
5
1
:
v
1 Introduction
i
X
r Let φ be a flow on X = Rd endowed with the usual topology, i.e., φ: R X X
a
× →
is continuous, and φ(0,x) = x as well as φ s,φ(t,x) = φ(s+t,x) for all x X
∈
and s,t R. Denoting the homeomorphismx φ(t,x) of X simply by φ and the
∈ (cid:0) 7→ (cid:1) t
spaceofalllinearmapsA:X X by (X), asusual,callthe flowφlinear if each
→ L
φ is linear, that is, φ (X) for every t R. Given a linear flow φ on X, fix any
t t
∈L ∈
linear functional H : (X) R and consider the function H(φ ). The main goal
•
L →
of this article is to completely describe the distribution of numerical values for the
real-valued functions thus generated.
To see why this distribution may be of interest, recall that throughout science
andengineering,flowsonthe phasespaceX =Rd areoftenusedto providemodels
for real-worlds processes; e.g., see [1]. From a scientist’s or engineer’s perspective,
itmaynotbedesirableorevenpossibletoobserveaflowφinitsentirety,especially
if d is large. Rather, what matters is the behaviour of certain functions (“signals”)
distilled from φ. Adopting terminology used similarly in e.g. quantum mechanics
and ergodic theory [9, 18], call any function h : X R an observable (on X).
→
1
With this, what really matters from a scientist’s or engineer’s point of view are
properties of signals h φ( ,x) for specific observables h and points x X that
• ∈
are relevant to the process being modelled by φ. In the case of linear flows, a
(cid:0) (cid:1)
specialroleis naturallyplayedby linear observables. Note that ifφ andh both are
linear then h φ(t,x) H(φ ), where H : (X) R is the linear functional with
t
≡ L →
H(A) = h(Ax) for all A (X). Given any linear flow φ on X, it makes sense,
(cid:0) (cid:1) ∈ L
therefore, to more generally consider signals H(φ ) where H : (X) R is any
•
L →
linear functional; by a slight abuse of terminology, such functionals will henceforth
be referred to as linear observables on (X) as well.
L
What,ifanything,canbesaidaboutthedistributionofvaluesforsignalsH(φ ),
(cid:0) (cid:1) •
whereφandH arealinearflowonX andalinearobservableon (X),respectively?
L
As indicated below and demonstrated rigorouslythrough the results of this article,
fortheoverwhelmingmajorityoflinearflowsthisquestionhasasurprisinglysimple,
though perhaps somewhat counter-intuitive answer: Except for the trivial case of
H(φ ) = 0, that is, H(φ ) = 0 for all t R, the values of H(φ ) always exhibit
• t •
∈
one and the same distribution, regardless of d, φ and H. As it turns out, this
distinguisheddistributionisnothingotherthanBenford’sLaw (BL),thelogarithmic
law for significant digits.
Withinthestudyof(digitsof)numericaldatageneratedbydynamicalprocesses
— a classical subject that continues to attract interest from disciplines as diverse
as ergodic and number theory [2, 10, 12, 21, 24], analysis [8, 25], and statistics
[14, 17, 26] — the astounding ubiquity of BL is a recurring, popular theme. The
most well-known special case of BL is the so-called (decimal) first-digit law which
asserts that
P(leading digit =ℓ)=log 1+ℓ−1 ℓ=1,...,9, (1.1)
10 10 ∀
(cid:0) (cid:1)
where leading digit refers to the leading (or first significant) decimal digit, and
10
log is the base-10 logarithm (see Section 2 for rigorous definitions); for example,
10
theleadingdecimaldigitofe=2.718is2,whereastheleadingdigitof ee = 15.15
− −
is 1. Note that (1.1) is heavily skewed towards the smaller digits: For instance,
the leading decimal digit is almost six times more likely to equal 1 (probability
log 2 = 30.10%) than to equal 9 (probability 1 log 9 = 4.57%). Ever since
10 − 10
first recorded by Newcomb [28] in 1881 and re-discovered by Benford [3] in 1938,
examplesofdataandsystemsconformingto(1.1)inoneformoranotherhavebeen
discussedextensively,notablyforreal-lifedata(e.g.[15,29])aswellasinstochastic
(e.g.[31])anddeterministicprocesses(e.g.theLorenzflow[33]andcertainunimodal
maps [7, 32]). As of this writing, an online database [4] devoted exclusively to BL
lists more than 800 references.
Given any (Borel) measurable function f : R+ R, arguably the simplest and
→
most natural notion of f conforming to (1.1) is to require that
λ( t T :leading digit f(t)=ℓ )
lim { ≤ 10 } =log 1+ℓ−1 ℓ=1,...,9;
T→+∞ T 10 ∀
(1.2)
(cid:0) (cid:1)
2
hereandthroughout,λdenotesLebesguemeasureonR+,oronpartsthereof. With
this,thecentralquestionstudiedhereinisthis: Does(1.2)holdforf =H(φ )where
•
φisalinearflowonX andH isanylinearobservableon (X)? Severalattemptsto
L
answerthisquestionarerecordedintheliterature;e.g.,see[5,20,27,33]. Allthese
attempts, however, seem to have led only to sufficient conditions for (1.2) that are
either restrictive or complicated to state. In contrast, Theorem 3.2 below, one of
the mainresultsofthis article,providesa simple necessary and sufficient condition
for every non-trivial signal f = H(φ ) to satisfy (1.2), and in fact to conform to
•
BL in aneven strongersense. All results in the literature alluded to earlierare but
simple special cases of this theorem.
To see why it is plausible for signals f = H(φ ) to satisfy (1.2), pick any real
•
number α = 0 and consider as an extremely simple but also quite compelling ex-
6
ample the function f(t) = eαt. Obviously, x = f is a solution of x˙ = αx, and
f(t) φ (R1) for the linear flow generated by this differential equation. A
t
≡ ∈ L
short elementary calculation shows that, for all T >0 and 1 ℓ 9,
≤ ≤
λ( t T :leading digit eαt =ℓ ) 1
{ ≤ 10 } log 1+ℓ−1 < , (1.3)
T − 10 αT
(cid:12) (cid:12) | |
(cid:12) (cid:0) (cid:1)(cid:12)
and hence(cid:12)(cid:12)(1.2) holds for f(t)=eαt whenever α=0. (Trivially(cid:12)(cid:12), it does not hold if
6
α=0.) However,already for the linear flow φ on R2 generated by
α β
x˙ = − x,
" β α #
withα,β Randβ >0,abrute-forcecalculationisoflittleuseindecidingwhether
∈
allnon-trivialsignalsf =H(φ ) satisfy (1.2). Theorem3.2 showsthat indeed they
•
do, provided that απ/(βln10) is irrational; see Example 3.4.
This article is organized as follows. Section 2 introduces the formal definitions
and analytic tools required for the analysis. In Section 3, the main results char-
acterizing conformance to BL for linear flows are stated and proved, based upon a
tailor-made notion of exponential nonresonance (Definition 2.9). Several examples
arepresentedinordertoillustratethisnotionaswellasthe mainresults. Section4
establishesthefactthat,assuggestedbythesimpleexamplesintheprecedingpara-
graph, exponential nonresonance, and hence conformance to BL as well, is generic
for linear flows on Rd. Given the widespread use of linear differential equations
as models throughout the sciences, the results of this article may contribute to a
betterunderstandingof,anddeeperappreciationfor,BLanditsapplicationsacross
a wide range of disciplines.
2 Definitions and tools
The following, mostly standard notation and terminology is used throughout. The
symbols N, Z+, Z, Q, R+, R, and C denote the sets of, respectively, positive in-
teger, non-negative integer, integer, rational, non-negative real, real, and complex
3
numbers, and ∅ is the empty set. Recall that Lebesgue measure on R+ or subsets
thereofiswrittensimply asλ. Foreachintegerb 2,the logarithmbaseb ofx>0
≥
is denoted log x, and lnx is the natural logarithm (base e) of x; for convenience,
b
let log 0 := 0 for every b, and ln0 := 0. Given any x R, the largest integer
b ∈
not larger than x is symbolized by x . The real part, imaginary part, complex
⌊ ⌋
conjugate,and absolute value (modulus) of any z C is z, z, z, and z , respec-
∈ ℜ ℑ | |
tively. For each z C 0 , there exists a unique number π < argz π with
∈ \{ } − ≤
z = z eıargz. Given any w C and C, define w+ := w+z : z and
| | ∈ Z ⊂ Z { ∈ Z}
w := wz :z . Thus with the unit circle S:= z C: z =1 , for example,
Z { ∈Z} { ∈ | | }
w+S = z C : z w = 1 and wS = z C : z = w for each w C. The
{ ∈ | − | } { ∈ | | | |} ∈
cardinality (number of elements) of any finite set C is # .
Z ⊂ Z
Recall throughout that b is an integer with b 2, informally referred to as a
≥
base. Given a base b and any x = 0, there exists a unique real number S (x) with
b
6
1 S (x) < b and a unique integer k such that x = S (x)bk. The number S (x)
b b b
≤ | |
is the significand or mantissa (base b) of x; for convenience, define S (0) := 0 for
b
every base b. The integer S (x) is the first significant digit (base b) of x; note
b
⌊ ⌋
that S (x) 1,...,b 1 whenever x=0.
b
⌊ ⌋∈{ − } 6
In this article, conformance to BL for real-valued functions, specifically for sig-
nals generated by linear flows, is studied via the following definition.
Definition 2.1. Let b N 1 . A (Borel) measurable function f : R+ R is a
∈ \{ } →
b-Benford function, or b-Benford for short, if
λ t T :S f(t) s
b
lim ≤ ≤ =log s s [1,b).
T→+∞ T b ∀ ∈
(cid:0)(cid:8) (cid:0) (cid:1) (cid:9)(cid:1)
Thefunction f isaBenford function,orsimply Benford, ifitis b-Benfordforevery
b N 1 .
∈ \{ }
Note that (1.2) holds whenever f is 10-Benford. The converse is not true in
generalsince, for instance, the (piecewise constant) function S (2•) only attains
10
⌊ ⌋
thevalues1,...,9andhenceclearlyisnot10-Benford,yet(1.3)withα=ln2shows
that it does satisfy (1.2).
The subsequent analysis of the Benford property for signals generatedby linear
flows is greatly facilitated by a few basic facts from the theory of uniform distribu-
tion, reviewed here for the reader’s convenience; e.g., see [16, 23] for authoritative
accounts of the subject. Throughout, the symbol d denotes a positive integer,
usually unspecified or clear from the context. The d-dimensional torus Rd/Zd is
symbolized by Td, its elements being represented as x =x+Zd with x Rd; for
h i ∈
simplicity write T instead of T1. Endow Td with its usual (quotient) topology and
σ-algebra (Td) of Borel sets, and let (Td) be the set of all probability measures
B P
on Td, (Td) . Denote the Haar (probability) measure of the compact Abelian
B
group Td by λ . Call a set T an arc if = := x :x for some in-
(cid:0) (cid:1)Td J ⊂ J hIi {h i ∈I}
terval R. With this, a (Borel) measurable function f :R+ R is continuously
I ⊂ →
uniformly distributed modulo one, henceforth abbreviated c.u.d. mod 1, if
λ t T : f(t)
lim ≤ h i∈J =λ ( ) for every arc T.
T→+∞ T
T J J ⊂
(cid:0)(cid:8) (cid:9)(cid:1)
4
Equivalently, lim 1 T F f(t) dt = F dλ for every continuous (or just
T→+∞ T 0 h i T T
Riemann integrable) function F :T C. In particular, therefore, if the function f
R (cid:0) →(cid:1) R
is c.u.d. mod 1 then lim 1 T e2πıkf(t)dt = 0 for every k Z 0 , and the
T→+∞ T 0 ∈ \{ }
converse is also true (Weyl’s criterion [23, Thm.I.9.2]).
R
The importance of uniform distribution concepts for the present article stems
from the following fact which, though very simple, is nevertheless fundamental for
all that follows.
Proposition 2.2.[13, Thm.1] Let b N 1 . A measurable function f :R+ R
∈ \{ } →
is b-Benford if and only if the function log f is c.u.d. mod 1.
b| |
In order to enable the effective application of Proposition 2.2, a few basic facts
from the theory of uniform distribution are re-stated here. In this context, the
following discrete-time analogue of continuous uniform distribution is also useful:
A sequence (x ) ofrealnumbers,by definition, is uniformly distributed modulo one
n
(u.d. mod 1) if the (piecewise constant) function f = x is c.u.d. mod 1, or
1+⌊•⌋
equivalently, if
# n N : x
limN→∞ { ≤ h ni∈J} =λT( ) for every arc T.
N J J ⊂
Lemma2.3. Foreachmeasurablefunctionf :R+ Rthefollowingareequivalent:
→
(i) f is c.u.d. mod 1;
(ii) If g : R+ R is measurable and lim g(t) f(t) exists (in R) then g
t→+∞
→ −
is c.u.d. mod 1;
(cid:0) (cid:1)
(iii) kf is c.u.d. mod 1 for every k Z 0 ;
∈ \{ }
(iv) f +αlnt is c.u.d. mod 1 for every α R.
∈
Proof. Clearly,(ii),(iii),and(iv)eachimplies(i),andtheconverseis[23,Exc.I.9.4],
[23, Exc.I.9.6], and [5, Lem.2.8], respectively.
The next result is a slight generalization of [23, Thm.I.9.6(b)].
Lemma 2.4. Let the function f :R+ R be measurable, and δ >0. If, for some
0
→
measurable, bounded F :T C and z C,
→ ∈
1 N
lim F f(nδ) =z for almost all 0<δ <δ ,
N→∞ 0
N n=1 h i
X (cid:0) (cid:1)
then also
1 T
lim F f(t) dt=z. (2.1)
T→+∞
T h i
Z0
(cid:0) (cid:1)
In particular, if the sequence f(nδ) is u.d. mod 1 for almost all 0 < δ < δ then
0
f is c.u.d. mod 1.
(cid:0) (cid:1)
5
Proof. For each n N, let z = δ0n F f(t) dt. By the Dominated Conver-
∈ n δ0(n−1) h i
gence Theorem,
R (cid:0) (cid:1)
1 δ0 1 N
lim F f(nδ) dδ =z.
N→∞
δ0 Z0 N n=1 h i
X (cid:0) (cid:1)
On the other hand,
1 δ0 1 N 1 N 1 δ0n
F f(nδ) dδ = F f(t) dt
δ0 Z0 N n=1 h i δ0N n=1 nZ0 h i
X (cid:0) (cid:1) X (cid:0) (cid:1)
1 N 1 n
= z ,
ℓ
δ0N n=1 n ℓ=1
X X
and since the sequence (z ) is bounded, a well-known Tauberian theorem [19,
n
Thm.92] implies that
1 N 1 δ0N
z =lim z =lim F f(t) dt
N→∞ n N→∞
δ0N n=1 δ0N Z0 h i
X (cid:0) (cid:1)
1 T
=lim F f(t) dt.
T→+∞
T h i
Z0
(cid:0) (cid:1)
The second assertion now follows immediately by considering specifically the func-
tions F( x )=e2πıkx for k Z, together with Weyl’s criterion.
h i ∈
The following result pertains to very particular functions that map Td into T;
suchfunctionswillappearnaturallyinthenextsection. Concretely,letp ,...,p
1 d
∈
Z and α R 0 , and consider the function
∈ \{ }
Td T,
P : →
u
( x p1x1+...+pdxd+αln u1cos(2πx1)+...+udcos(2πxd) ;
h i 7→ | |
(cid:10) (cid:11)
here u Rd 0 may be thought of as a parameter. (Recall the convention that
∈ \{ }
ln0=0.) Note that P is measurable (in fact, differentiable λ -a.e.), and so each
u Td
µ (Td) induces a well-defined element µ P−1 of (T), via µ P−1(B) :=
∈ P ◦ u P ◦ u
µ P−1(B) for all B (T). It is easy to see that µ P−1 is absolutely continuous
u ∈B ◦ u
(w.r.t. λ ) whenever µ is absolutely continuous (w.r.t. λ ). For the purpose of
(cid:0) T(cid:1) Td
this work, only the case µ = λ is of further interest. Observe that λ P−1 is
Td Td ◦ u
equivalent to (i.e., has the same nullsets as) λ . Moreover,for (T) endowed with
T
P
the topology of weak convergence, the Dominated Convergence Theorem implies
that the (T)-valued function u λ P−1 is continuous on Rd 0 , for any
P 7→ Td ◦ u \{ }
fixed p ,...,p Z and α R 0 . The arguments in [6, Sec.5] show that this
1 d
∈ ∈ \{ }
function is non-constant, as might be expected.
Proposition2.5.[6,Thm.5.4]Givenp ,...,p Z,α R 0 ,andanyν (T),
1 d
∈ ∈ \{ } ∈P
there exists u Rd 0 such that λ P−1 =ν.
∈ \{ } Td ◦ u 6
Remark 2.6. Specifically for the case ν = λ , it has been conjectured in [6] that
T
λ P−1 =λ (if and) only if u =0, and hence λ P−1 =λ for most
Td ◦ u T j:pj6=0 j Td ◦ u 6 T
u Rd 0 .
∈ \{ } Q
6
The remainder of this section reviews tools and terminology concerning certain
elementary number-theoretical properties of sets C. Specifically, denote by
Z ⊂
span the smallest subspace of C (over Q) containing ; equivalently, if = ∅
QZ Z Z 6
then span is the set of all finite rational linear combinations of elements of ,
QZ Z
i.e.,
span = ρ z +...+ρ z :n N,ρ ,...,ρ Q,z ,...,z ;
QZ 1 1 n n ∈ 1 n ∈ 1 n ∈Z
note that span ∅(cid:8)= 0 . With this terminology, recall that z ,...,z C(cid:9) are Q-
Q { } 1 L ∈
independent ifspan z ,...,z is L-dimensional,orequivalently if L p z =0
Q{ 1 L} ℓ=1 ℓ ℓ
with integers p ,...,p implies p = ... = p = 0. The notion of Q-independence
1 L 1 L P
is crucial for the distribution mod 1 of certain sequences and functions, and hence,
viaProposition2.2,alsoforthestudy ofBL.Asimplebutusefulfactinthis regard
is as follows.
Proposition 2.7.[6,Lem.2.6] Let ϑ ,ϑ ,...,ϑ R, and assume that the function
0 1 d
∈
F :Td Ciscontinuous,andnon-zeroλ -almosteverywhere. Ifthed+2numbers
Td
→
1,ϑ ,ϑ ,...,ϑ are Q-independent then the sequence
0 1 d
nϑ +αlnn+βln F (nϑ ,...,nϑ ) +z
0 1 d n
h i
is u.d. mod 1 for(cid:16)every α,β R and e(cid:12)ve(cid:0)ry sequence (z ) i(cid:1)n C w(cid:12)i(cid:17)th lim z =0.
(cid:12) n (cid:12) n→∞ n
∈
The following definitions of nonresonance and exponential nonresonance have
beenintroducedin[6]and[7],respectively. Aswillbecomeclearinthenextsection,
they owe their specific form to Propositions 2.2, 2.5, and 2.7.
Definition 2.8. Let b N 1 . A non-empty set C with z = r for some
∈ \{ } Z ⊂ | |
r >0 and all z , i.e. rS, is b-nonresonant if the associated set
∈Z Z ⊂
argz argw
∆ := 1+ − :z,w R
Z
2π ∈Z ⊂
(cid:26) (cid:27)
has the following two properties:
(i) ∆ Q= 1 ;
Z
∩ { }
(ii) log r span ∆ .
b 6∈ Q Z
An arbitrary set C is b-nonresonantif, for every r >0, the set rS is either
Z ⊂ Z∩
b-nonresonantor empty; otherwise, is b-resonant.
Z
Definition 2.9. Let b N 1 . A set C is exponentially b-nonresonant if
∈ \{ } Z ⊂
the set etZ := etz : z is b-nonresonant for some t R+; otherwise, is
{ ∈ Z} ∈ Z
exponentially b-resonant.
Example 2.10. The empty set ∅ is b-nonresonant and exponentially b-resonant
for every b. The singleton z with z C is b-nonresonant if and only if either
{ } ∈
z = 0 or log z Q, and it is exponentially b-nonresonant precisely if z = 0.
b| | 6∈ ℜ 6
Similarly, any set z,z with z C R is b-nonresonant if and only if 1, log z
{ } ∈ \ b| |
and 1 argz are Q-independent, and it is exponentially b-nonresonant precisely if
2π
zπ/( zlnb) Q.
ℜ ℑ 6∈
7
Notethatif is(exponentially)b-nonresonantthensoarethesets :=( 1)
Z −Z − Z
and := z : z , as well as every . Also, for each n N the set
Z { ∈ Z} W ⊂ Z ∈
n := zn : z is b-nonresonant whenever is. The converse fails since, for
Z { ∈ Z} Z
instance, = e,e is b-resonant whereas 2 = e2 is b-nonresonant. Similarly,
Z {− } Z { }
if is exponentially b-nonresonant then so is t for all t R 0 . On the other
Z Z ∈ \{ }
hand,aset iscertainlyb-resonantif S=∅,anditisexponentiallyb-resonant
Z Z∩ 6
whenever ıR=∅.
Z ∩ 6
The following simple observation establishes an alternative description of expo-
nential b-nonresonance. Recall that a set is countable if it is either finite (possibly
empty) or countably infinite.
Lemma 2.11. Let b N 1 . Assume that the set C is countable and
∈ \ { } Z ⊂
symmetric w.r.t. the real axis, i.e., = . Then the following are equivalent:
Z Z
(i) is exponentially b-nonresonant;
Z
(ii) For every z ,
∈Z
lnb
z span w :w , w= z . (2.2)
ℜ 6∈ Q π ℑ ∈Z ℜ ℜ
(cid:26) (cid:27)
Moreover, if (i) and (ii) hold then the set t R+ :etZ is b-resonant is countable.
{ ∈ }
Proof. Toshow(i) (ii),supposethereexistdifferentelementsw ,...,w of with
1 L
⇒ Z
w =...= w , as well as p ,...,p Z and q N such that
1 L 1 L
ℜ ℜ ∈ ∈
L pℓlnb
w = w .
1 ℓ
ℜ ℓ=1 q π ℑ
X
Pick any t>0, let r :=etℜw1, and note that
t w1 L pℓt wℓ
log r = ℜ = ℑ .
b lnb ℓ=1 q π
X
On the other hand, since is symmetric w.r.t. the real axis, and since argetwℓ
Z
differs from t w by an integer multiple of 2π,
ℓ
ℑ
t w
ℓ
spanQ∆etZ∩rS ⊃spanQ {1}∪ ℑπ :ℓ=1,...,L .
(cid:18) (cid:26) (cid:27)(cid:19)
Thus logbr ∈ spanQ∆etZ∩rS, showing that etZ is b-resonant for all t > 0. Since
clearlye0Z = 1 is b-resonantaswell, is exponentiallyb-resonant,contradicting
{ } Z
(i). Hence (i) (ii); note that the countability of has not been used here.
⇒ Z
To establish the reverseimplication (ii) (i), suppose the set is exponentially
⇒ Z
b-resonant. In this case, for everyt>0 there exists r =r(t)>0 such that etZ rS
∩
isb-resonant,andsoeither∆etZ∩rS∩Q6={1}orlogbr ∈spanQ∆etZ∩rS,orboth. In
thefirstcase,thereexistelementsw =w (t)andw =w (t)of with w = w
1 1 2 2 1 2
Z ℜ ℜ
but w =w such that t( w w ) πQ 0 . In particular, therefore,
1 2 1 2
6 ℑ −ℑ ∈ \{ }
π
t Q =: Ω . (2.3)
1
∈ z∈Z w∈Z\{z}:ℜw=ℜz w z
ℑ −ℑ
[ [
8
Inthesecondcase,forsomepositiveintegerL=L(t)andsomew (t),...,w (t)
1 L
∈Z
with w =...= w =t−1lnr,
1 L
ℜ ℜ
t w t w
1 ℓ
logbr = ℜlnb ∈spanQ∆etZ∩rS ⊂spanQ {1}∪ ℑπ :ℓ=1,...,L .
(cid:18) (cid:26) (cid:27)(cid:19)
With the appropriate p (t),p (t),...,p (t) Z and q(t) N, therefore,
0 1 L
∈ ∈
L lnb
t q w p w =p lnb. (2.4)
1 ℓ ℓ 0
ℜ − ℓ=1 π ℑ
(cid:18) (cid:19)
X
Since is countable, the set Ω in (2.3) is countable as well. Consequently, if
1
Z Z
is exponentially b-resonant then (2.4) must hold for all but countably many t > 0.
Hencethereexistt >t >0suchthatL(t )=L(t ),w (t )=w (t )andsimilarly
2 1 2 1 ℓ 2 ℓ 1
p (t )=p (t ) for all ℓ=1,...,L, as well as q(t )=q(t ). This in turn implies
ℓ 2 ℓ 1 2 1
L pℓlnb
w = w ,
1 ℓ
ℜ ℓ=1 q π ℑ
X
which clearly contradicts (2.2). For countable , therefore, (ii) fails whenever (i)
Z
fails, that is, (ii) (i); note that the symmetry of has not been used here.
⇒ Z
Finally, if (i) and (ii) hold, and if etZ is b-resonantfor some t>0 then, as seen
in the previous paragraph,either t Ω or else, by (2.4),
1
∈
lnb lnb
span z w :w , w= z =: Ω .
t ∈ z∈Z Q {ℜ }∪ π ℑ ∈Z ℜ ℜ 2
(cid:18) (cid:26) (cid:27)(cid:19)
[
Since is countable, so are Ω and Ω , and hence t R+ : etZ is b-resonant is
1 2
Z { ∈ }
countable as well.
Remark2.12. ThesymmetryandcountabilityassumptionsareessentialinLemma
2.11. If is not symmetric w.r.t. the real axis then the implication (i) (ii) may
Z ⇒
fail,as is seene.g.for = 1+ıπ/ln10 whichis exponentially10-nonresonantby
Z { }
Example 2.10, yet does not satisfy (ii) for b = 10. Conversely, if is uncountable
Z
then (ii) (i) may fail. To see this, simply take =R 0 which satisfies (ii) for
⇒ Z \{ }
all b, and yet etZ is b-resonantfor every t R+.
∈
Deciding whether a set C is b-resonant may be difficult in practice, even
Z ⊂
if # = 2. For example, it is unknown whether z C : z2 +2z +3 = 0 is
Z { ∈ }
10-resonant;see [7, Ex.7.27]. In many situations of practical interest, the situation
regarding exponential b-resonance is much simpler. Recall that a number z C is
∈
algebraic (over Q) if it is the root of some non-constant polynomial with integer
coefficients.
Lemma 2.13. Let b N 1 . Assume every element of C is algebraic. Then
∈ \{ } Z ⊂
is exponentially b-nonresonant if and only if ıR=∅.
Z Z∩
Proof. The“onlyif”partisobvioussince,asseenearlier, ıR=∅alwaysrenders
Z∩ 6
theset exponentiallyb-resonant. Toprovethe“if”part,supposethat ıR=∅
Z Z∩
yet is exponentially b-resonant. Since all of its elements are algebraic, the set
Z Z
9
is countable. By Lemma 2.11 there exist z ,...,z with z = ... = z , as
1 L 1 L
∈ Z ℜ ℜ
well as p ,...,p Z and q N such that
1 L
∈ ∈
L pℓlnb
z = z .
1 ℓ
ℜ ℓ=1 q π ℑ
X
(Recallthat the proofof the implication(ii) (i) in that lemma does notrequire
⇒ Z
to be symmetric w.r.t. the real axis.) Since z =0, it follows that
1
ℜ 6
π L pℓ zℓ
= ℑ ,
lnb ℓ=1 q z1
ℜ
X
which in turn implies that π/lnb is algebraic. However,by the Gel’fond–Schneider
Theorem[34,Thm.1.4],thenumberπ/lnbisnot algebraicforanyb N 1 . This
∈ \{ }
contradiction shows that cannot be exponentially b-resonantif ıR=∅.
Z Z ∩
3 Characterizing BL for linear flows on Rd
Let φ be a linear flow on X = Rd. Recall that there exists a unique A (X)
φ
∈ L
such that φt = etAφ for all t R. (The linear map Aφ is sometimes referred
∈
to as the generator of φ.) In fact, with I (X) denoting the identity map,
X
∈ L
A = lim t−1(φ I ) = dφ , and φ is simply the flow generated by the
φ t→0 t − X dt t|t=0
(autonomous) linear differential equation x˙ = A x. Since conversely (t,x) etAx
φ
7→
defines, for each A (X), a linear flow on X with generator A, there is a one-to-
∈L
onecorrespondencebetweenthefamilyofalllinearflowsonX andthespace (X).
L
Thus it makes sense to define the spectrum of φ as
σ(φ):=σ(A )= z C:z is an eigenvalue of A .
φ φ
{ ∈ }
Note that σ(φ) C is non-empty, countable (in fact, finite with #σ(φ) d) and
⊂ ≤
symmetric w.r.t. the real axis.
Recallthat the symbol H is used throughoutto denote a linear observable(i.e.,
a linear functional) on (X). For convenience, let (X) be the space of all such
L O
observables,i.e., (X)issimplythedualof (X),endowedwiththeusualtopology.
O L
The following is a basic linear algebra observation.
Lemma 3.1. Let φ be a linear flow on X. Given any non-empty set σ(φ) and
Z ⊂
any vector u RZ, there exists H (X) such that
∈ ∈O
H(φ )= etℜzu cos(t z) t R. (3.1)
t z
z∈Z ℑ ∀ ∈
X
Proof. For every real z , pick v X 0 such that A v = zv and let
z φ z z
∈ Z ∈ \ { }
v :=v . For every non-real z , pick v ,v X 0 such that
z z z z
∈Z ∈ \{ }
A v =v z v z, A v =v z+v z;
e φ z zℜ − zℑ eφ z zℑ zℜ
note that v ,v are linearly independent. With this, for each z ,
z z e e e ∈Z
φ v =etℜz v cos(t z) v sin(t z)
e t z z ℑ − z ℑ t R. (3.2)
φtvz =etℜz(cid:0)vzsin(t z)+vzcos(t z)(cid:1) ∀ ∈
ℑ e ℑ
(cid:0) (cid:1)
e e
10