C.3 Some properties of Fourier transformation
In this section some basic properties of the Fourier transformation for functions and functionals are presented, which are used throughout the work.
Proposition C.3.1. Let q,r ∈Z, then the Bochner transformδˆ︃qZ+r ofδqZ+r exists and is given byδˆ︃qZ+r= 1qe−rδ1
qZq, where ey(x)≔e2πixy.
Proof. Thatδˆ︃qZ+rexists is due toδqZ+r =δqZ∗δr∈SFP(Z). Letϕ∈ C([0,1)), then
⟨δqZ+r,ˆ︁ϕ⟩=∑︂
z∈Z
ˆ︁ϕ(qz+r)
=∑︂
z∈Z
∫︂ 1 0
ϕ(y)e−2πi(qz+r)ydy= ∑︂
z∈Z
∫︂ 1 0
ϕ(y)e−2πirye−2πiqzydy
=∑︂
z∈Z q−1
∑︂
k=0
∫︂ k+q1
k q
ϕ(y+k/q)e−2πir(y+qk)e−2πiqz(y+qk)dy
=∑︂
z∈Z q−1
∑︂
k=0
∫︂ k+q1
k q
ϕ(y+k/q)e−2πir(y+qk)e−2πiqzydy
=∑︂
z∈Z
1 q
∫︂ 1 0
q−1
∑︂
k=0
ϕ(︂w+k
q
)︂e−2πirw+kq e−2πizwdw
=∑︂
z∈Z q−1
∑︂
k=0
1 q
∫︂ 1 0
ψ(w)e−2πizw dw
=1 q
∑︂
z∈Z
ˆ︁ψ(z)e0= 1
qψ(0)= 1 q
q−1
∑︂
k=0
ϕ(︂k
q
)︂e−2πirkq = ⟨︃
q−1e−r(k/q)δ1
qZq, ϕ⟩︃
, whereψ(w)≔∑︁q−1
k=0ϕ(︂w+k
q
)︂e−2πirw+kq .
Lemma C.3.2. Supposeη∈CFP(Z)and denote by sr: Z→ Zthe map x↦→ x+r for all r∈Z. Thenη◦srhas a Bochner transform given by erˆ︁η, where er(x)= e2πirx for all x∈[0,1).
Proof. Let f ∈CP(Z), then
⟨η◦sr, f⟩=⟨η, f ◦s−r⟩= ⟨ˆ︁η,(f ◦s−r)∨⟩
=⟨ˆ︁η,er f∨⟩=⟨erˆ︁η, f∨⟩, as for allx∈[0,1]
(f ◦s−r)∨(x)= ∫︂
f(y−r)e2πixydm(y)= ∫︂
f(y)e2πix(y+r)dm(y)= er(x)f∨(x).
Proposition C.3.3(Poisson summation formula). Let f ∈ Cc(R,C), then
∑︂
z∈Z
ˆ︁f(z)= ∑︂
z∈Z
f(z).
Proof. Defineg: [0,1) → C to be g(x) ≔ ∑︁
z∈Z f(x+z), which is well-defined, since f has compact support.
∑︂
z∈Z
f(0+z)=g(0)
=∑︂
z∈Z
ˆ︁g(z)e2πiz0
=∑︂
z∈Z
∫︂ 1 0
g(y)e−2πizydm(y)
=∑︂
z∈Z
∫︂ 1 0
∑︂
m∈Z
f(y+m)e−2πiz(y+m)dm(y)
=∑︂
z∈Z
∑︂
m∈Z
∫︂ m+1 m
f(y)e−2πizy dm(y)
=∑︂
z∈Z
∫︂
R
f(y)e−2πizy dm(y)= ∑︂
z∈Z
ˆ︁f(z).
Where the dominated convergence theorem holds, as ∫︁ ∑︁
m∈Z|1[m,m+1)(y) f(y + m)| dm|[0,1)(y) ≤ ∫︁
⌈supp(f)⌉ sup|f| dm|[0,1) < ∞, which allows to interchange
summation and integration.
Danksagung
Ich m̈ochte mich bei Johannes daf̈ur bedanken, dass er mich das erste Jahr ̈uber begleitet hat. Außerdem danke ich Christina und Lars f̈ur die vielen anregenden Diskussionen. Dies schließt Hendrik mit ein, der mich dar̈uber hinaus besonders in der Abschlussphase meiner Arbeit untersẗutzt hat. Auch m̈ochte ich gerne Mo-ritz und Konstantin danken, die ich noch am Ende meiner Dissertation kennen lernen durfte. Großer Dank gilt Kathryn, die mich in allen Fragen rund um die Universiẗat betreut hat. Des Weiteren m̈ochte ich Tony f̈ur viele fruchtbringende Diskussionen und die Zeit danken, die er mir zu Verf̈ugung gestellt hat. Besonde-rer Dank gilt noch Malte Steffens, der mich die ganze Zeit ̈uber begleitet hat und ohne den die Quest sicherlich ganz anders verlaufen ẅare. Abschließend m̈ochte ich mich noch bei Nicolae Strungaru bedanken, den ich ẅahrend Oberwolfach n̈aher kennen lernen durfte. Schlussendlich m̈ochte ich noch besonders Marc dan-ken, der mir diesen Pfad erst er̈offnet hat und Daniel Lenz f̈ur seine bereitwillige Untersẗutzung.
183
Nomenclature
∆ Parameter space for (intermediate)β-transformations
δC ∑︁
c∈Cδc, whereC ⊆ Xis discrete and at most countable δc Dirac point mass atc, wherec∈X
η(n,r)(z) For an η ∈ C′c(Z) given by η(zn) if it exists an zn ∈ Z such that z=zn+rand 0 otherwise, see Definition 8.4.4
γ TheT-discontinuity of aβ-transformation,γ= dn−k+1 =(1−α)/β γ, γηy, γT Autocorrelation of the return time comb ηy with respect to T and
reference pointy, given by∑︁
z∈ZΞ(T, ν)(z)δz, see Theorem 6.3.2 γu Autocorrelation of a wordu, see Definition 6.6.8
κlp/q A finite word encoding rational rotation, see Definition 2.3.7 µ A Borel measure or functional onCc
µ(−·) A↦→µ(−A) for allA∈B(G)
ν A Borel measure or functional onCc
ωlj A finite word associated with a rigid rotation, see Definition 2.3.2 φ(m, j,r,t) Given by (jqm+1+qm)tr∑︁am+2
l=j (lqm+1+qm)−t, see (3.18)
Φ1 Operator used in the spectral decomposition of the Perron-Frobenius operator, see Theorem 6.5.7
φi A function given byϕm,im ◦. . .◦ϕ0,i0 form∈N, see Theorem 8.4.10 Ψ Operator used in the spectral decomposition of the Perron-Frobenius
operator, see Theorem 6.5.7
ψ(r) Given by sup{ψw(r) : w∈X}, see (3.10)
ψw(r) Given by lim supv→wdξ,t(w,v)/dt(w,v)r, see (3.10)
ψx,n(r) Given bydξ,t(x,S|Ln|(y))/dt(x,S|Ln|(y))r, see Definition 3.3.7
ψ(x,nj)(r) Given bydξ,t(x,S(a2(n+1)−j+1)|Ln+1|(y))/dt(x,S(a2(n+1)−j+1)|Ln+1|(y))r; (3.16) ψy,n(r) Given bydξ,t(S|Rn|(x),y)/dt(S|Rn|(x),y)r, see Definition 3.3.7
185
ψ(i)y,n(r) Given bydξ,t(S(a2(n+1)−1−i+1)|Rn|(x),y)/dt(S(a2(n+1)−1−i+1)|Rn|(x),y)r; (3.16) ρ Substitution which maps 0↦→ 01,1↦→1
ηy Return time measure, see Definition 6.3.1
Σ A finite set, often called alphabet and its elements are often called letters
σ A substitution as an element of the setQ Σ∗ Set of finite words for a finite alphabetΣ
σL(T) The spectrum of and operatorT, see Section 6.1.1 τ Substitution which maps 0↦→ 0,1↦→10
τTM Substitution which maps 0↦→ 01,1↦→10 θ Substitution which maps 0↦→ 1,1↦→0
Θα The setΘα∩Θαof well-approximable numbers ofα-type
ϕi,j The function ιi ◦Tini−1−j from Ii,j → [0,1] where 1 ≤ j ≤ ni, see Theorem 8.4.10, Lemma 8.3.3
ϱ Spectral (return) measure of the Thue-Morse substitution, see Propo-sition 8.5.16
ϱf Spectral measure with respect to a function f, see Definition 6.2.1 ϱα(t) A function onR given by 0, if t ≤ 1−1/α, by 1−(α−1)/(αt) if
1−1/α <t<1 and by 1/αift≥ 1, see Section 3.3.3 ϱmax Maximal spectral type of an operator, see Definition 6.4.2
Ξ(T, ν) Weight function of an autocorrelation of a return time comb, see Sec-tion 6.3
ζ A primitive substitution
ℓα Number to determine if a subshift isα-repulsive; Definition 2.2.14 C The set of complex numbers
N {0,1,2, . . .}
N+ {1,2,3, . . .}
Q The set of rational numbers R The set of real numbers Z The set of integers
Zq Group with one generator of periodq
C′c(X) The space of continuous linear functionals with respect to the vague-topology, see Appendix B.2.2
NOMENCLATURE 187 C(X) Space of continuous functions fromXtoC
Cb(X) Space of bounded continuous functions fromXtoC
Cc(X) Space of continuous functions fromXtoCwith compact support Jβc Jψforψ(y)=cy−β, whereβ >2 andc>0, see Definition 3.4.1 Jψ Theψ-Jarnı́k set, see Definition 3.4.1
L(u) The set containing all factors ofu
L(X) The language of the subshiftX, see Definition 2.2.7
Lk(ξ),Lk Approximation of yin the subshift of slope ξ by substitutions with the first 2k−1 continued fraction entries, see Definition 3.2.1 Rk(ξ),Rk Approximation of x in the subshift of slopeξ by substitutions with
the first 2kcontinued fraction entries, see Definition 3.2.1 T Topology of a space
h(ζ) The height of a primitive substitutionζ of constant lengthq; Defini-tion 6.6.10
M(X) The space of non-negative Borel measures onX M1(X) Space of Borel-probability measures onX
B,B(X) Borelσ-algebra generated by the topologyT, denoted byσ(T) BN The tail-σ-algebra ofT given by⋂︁
n∈NT−n(B), see Chapter 6
L(X,Y) Space of continuous linear operators, see Section 6.1 and Appendix B.2.1 L (B) Space of continuous linear operators on a Banach algebraBtoB, see
Section 6.1
M(X) The space of complex-valued regular Borel measures onX
Q A space of substitutions associated with rational rotations, see Defi-nition 8.5.2
[w]T The setTw−1([0,1)), see Section 8.1
[n]q,[n] An element ofZqw.r.t. the canonical projectionZ→(Z/qZ)
△ A△B≔(A\B)∪(B\A) whereA,B⊆ X 1A Mapsxto 1 if x∈Aand to 0 if x∉A
η∗ f The functiony↦→ ⟨η, f(y− ·)⟩, where f ∈ Cc(G) δc∗η For all f ∈ Cc(G) given by⟨δc∗η, f⟩=⟨η, δ−c∗ f⟩ δc∗ f A function given byx↦→ f(x−c)
⟨µ, f⟩ Given by∫︁
f dµ
⟨µ, f⟩ Given by∫︁
f dµ
⟨˜︁µ, f⟩ Given by∫︁
˜︁f dµ
⟨f,g⟩ Given by∫︁
f gdµ
⟨µ∗ν, f⟩ Given by∫︁
f(x+y) dµ(x) dν(y), f ∈ Cc
⌈x⌉ ⌈x⌉=z∈Zsuch that x−z+1∈[0,1), where x∈R
⌊x⌋ ⌊x⌋=z∈Zsuch that x−z∈[0,1), wherex∈R
µ∗η For all f ∈ Cc(G) given by⟨µ∗η, f⟩ = ⟨η, µ−∗ f⟩, whereµ−(A) = µ(−A) for allA∈B
µ∗ν Given byµ∗ν(A)≔∫︁
1A(x+y) dµ(x) dν(y), whereA∈B µ∗ f Given byµ∗ f(y)≔
∫︁ f(y−x) dµ(x), wherey∈G
Θα The set{ξ ∈ [0,1] : Aα(ξ) < ∞} of well-approximable numbers of α-type
A Closure of the setA
f(x) Given by the mappingx↦→ f(x), the complex conjugate of f(x) Θα The set {ξ ∈ [0,1] : 0 < Aα(ξ)} of well-approximable numbers of
α-type
ˆ︁γ,γˆ︂ηy,γˆ︁T Spectral return comb or Bochner transform of the autocorrelation γ, γηy andγT respectively, see Definition 6.3.3
˜︁f(x) Given by the mappingx↦→ f(−x) {x} The fractional part ofx
{x} The set containing the pointx A◦ Interior of the setA
f ∗g Convolution of functions, f ∗g(y)≔∫︁
f(x)·g(y−x) dmG(x) v∧w The longest common prefix ofvandw
BV The space of functions of bounded variation, see Proposition 6.5.1 Aα(ξ) For a subshift of slopeξgiven by lim supn→∞anq1−αn−1
CP(G) Continuous positive definite functions onG dimH Hausdorffdimension
Exact(β) Given byJβ1\⋃︁
n≥2,n∈N+Jβn/(n+1), see Definition 3.4.1 P(G) Positive definite functions onG
ResL(T) The resolvent of an operatorT, see Section 6.1.1 SCP(G) Span of continuous positive definite functions onG SFP(G) Span of positive definite functionals onG
NOMENCLATURE 189 var(f) Variation of a function f, see Section 6.5
A+ {x∈A:xis non-negative}, for some vector spaceV ⊇ A Aα,n See Definition 2.2.14
Cn A certain subset of∆, see Definition 8.2.1 Dℓ A certain subset of∆for a finite sequenceℓ ≔ (︁
(ki,ni))︁m
i=0, see Defi-nition 8.4.7
di Discontinuities ofTn, where T(x) = {βx+α}, (β, α) ∈ Cn and 1 ≤ i≤n+1, see Definition 8.2.2
dξ,t(v,w) The spectral metric ofvandwgiven by|v∨w|−t+∑︁
n>|v∨w|bn(v)n−t+
∑︁n>|v∨w|bn(w)n−t, see Definition 3.3.2
Dk,n Given by{(β, α)∈Cn :zk,zk+1 ∉(T(0),T(1))}, see Definition 8.2.8 ey For anyx∈Rgiven bye2πixy, domain is often [0,1)
f Usually a function of some space intoCorR f(−·) x↦→ f(−x) for allx∈G
fv(u) Frequency ofvinu, see Definition 2.2.2
G Locally compact, Hausdorff,σ-compact abelian group g Usually a function of some space intoCorR
h Normalised eigenfunction of the Perron-Frobenius operator, given in Theorem 6.5.7
h The Parry density for a dynamical system of aβ-transformation, see (8.2)
h Topological pressure
Ii Given by [Ti(0),Ti(1)) fori∈ {1, . . . ,n−1}and [0,Tn(1))∪[Tn(0),1) fori= n, see Lemma 8.3.3
m Lebesgue measure onRor [0,1)≅R/Z mG A fixed Haar measure onG
P The Perron-Frobenius operator, see Equation (6.12)
pn,qn Defined for a continued fraction such thatpn/qn =[0;a1, . . . ,an] pu,p The complexity function of a wordu, see Definition 2.2.4
q Often the denominator or length for a substition of constant length Q(n) See Definition 2.2.14
Qα Number to determine if a subshift isα-finite, see Definition 2.2.14 R Repetitive function for some subshift, see Definition 2.2.10
Rα Number to determine if a subshift isα-repetitive; Definition 2.2.12 S The left shift operator onΣNorσ∗
sb,a For anyx∈Rgiven bybx+a, domain is often [0,1) sb For anyx∈Rgiven bybx, hencesb= sb,0
T A map that is used for the dynamics of a system. That includes transformations,β-transformations and operators respectively T,Tβ,α The mappingx↦→ {βx+α}on [0,1) or [0,1], with 1↦→β−1+α Tα The mapx↦→ {x+α}, α > 0
Tµ Transfer operator of a measure µand a transformation T, see Sec-tion 6.5
UT Koopman operator of a transformationT, see Section 6.5
X A locally compact separable completely metrisable topological space X A subshift, that is a shift-invariant closed subset ofΣN
x Given by limk→∞Rktaken with respect todt, see Definition 3.2.1 X′ Space of continuous functionals, see Appendix B.2.1
Xζ Subshift of a primitive substitutionζ Xu Subshift of a wordu, usually minimal
y Given by limk→∞Lk taken with respect todt, see Definition 3.2.1 y The reference point of a return time comb, see Definition 6.3.1 zi Fori ∈ {1, . . . ,n}the fixed points ofTn, whereT(x) = {βx+α}and
(β, α)∈Cn, see Definition 8.2.5
Bibliography
[1] H. W. Alt, Lineare Funktionalanalysis, vol. 5 of Springer-Lehrbuch, Springer-Verlag Berlin Heidelberg, 1985.
[2] H. Amann andJ. Escher,Analysis II, Grundstudium Mathematik, Birkḧauser Verlag, 1999.
[3] L. Argabright andJ. Gil deLamadrid,Fourier analysis of unbounded mea-sures on locally compact abelian groups, American Mathematical Society, Providence, R.I., 1974. Memoirs of the American Mathematical Society, No. 145.
[4] M. Baake, F. G̈ahler,andU. Grimm,Spectral and topological properties of a family of generalised Thue-Morse sequences, J. Math. Phys., 53 (2012), pp. 032701, 24.
[5] M. Baake andU. Grimm,The singular continuous diffraction measure of the Thue-Morse chain, J. Phys. A, 41 (2008), pp. 422001, 6.
[6] M. Baake and U. Grimm, Kinematic diffraction from a mathematical view-point, Zeitschrift f̈ur Kristallographie Crystalline Materials, 226 (2011), pp. 711–725.
[7] , Aperiodic Order, Cambridge University Press, CPI Group Ltd, Croy-don, CR0 4YY, 2013.
[8] M. Baake and D. Lenz, Dynamical systems on translation bounded mea-sures: pure point dynamical and diffraction spectra, Ergodic Theory Dynam.
Systems, 24 (2004), pp. 1867–1893.
[9] M. Baake andR. V. Moody,Weighted Dirac combs with pure point diff rac-tion, J. Reine Angew. Math., 573 (2004), pp. 61–94.
[10] S. Beckus andF. Pogorzelski,Delone dynamical systems and spectral con-vergence, arxiv:1711.07644, (2017).
191
[11] J. V. Bellissard andJ. Pearson,Noncommutative riemannian geometry and diffusion on ultrametric cantor sets, J. Noncommut. Geom. (3) 3, (2009), pp. 447–480.
[12] D. Berend and C. Radin, Are there chaotic tilings?, Communications in Mathematical Physics, 152 (1993), pp. 215–219.
[13] C. Berg and G. Forst, Potential Theory on Locally Compact Abelian Groups, Ergebnisse der Mathamatik und ihrer Grenzbebiete, Band 87, Springer Verlag Berlin Heidelberg New York, 1975.
[14] V. Berthé and V. Delecroix, Beyond substitutive dynamical systems: S -adic expansions, in Numeration and substitution 2012, RIMS Kôkyûroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, 2014, pp. 81–123.
[15] N. H. Bingham, C. M. Goldie, andJ. L. Teugels, Regular variation, vol. 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1989.
[16] N. Bourbaki, Eléments de mathématique. XXI. Première partie: Les structures fondamentales de l’analyse. Livre VI: Intégration. Chapitre V:
Intégration des mesures, Actualités Sci. Ind., no. 1244, Hermann & Cie, Paris, 1956.
[17] , É léments de mathématique. XXV. Première partie. Livre VI:
Intégration. Chapitre 6: Intégration vectorielle, Actualités Sci. Ind. No.
1281, Hermann, Paris, 1959.
[18] , É léments de mathématique. Fascicule XXIX. Livre VI: Intégration.
Chapitre 7: Mesure de Haar. Chapitre 8: Convolution et représentations, Actualités Scientifiques et Industrielles, No. 1306, Hermann, Paris, 1963.
[19] , É léments de mathématique. Fasc. XIII. Livre VI: Intégration.
Chapitres 1, 2, 3 et 4: Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d’une mesure, Espaces Lp, Deuxième édition revue et augmentée. Actualités Scientifiques et Indus-trielles, No. 1175, Hermann, Paris, 1965.
[20] R. Bowen, Bernoulli maps of the interval, Israel J. Math., 28 (1977), pp. 161–168.
[21] Y. Bugeaud, Sets of exact approximation order by rational numbers, Math.
Ann., 327 (2003), pp. 171–190.
BIBLIOGRAPHY 193 [22] , Sets of exact approximation order by rational numbers. II, Unif.
Dis-trib. Theory, 3 (2008), pp. 9–20.
[23] A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
[24] A. Deitmar andS. Echterhoff, Principles of Harmonic Analysis, Universi-tytext, Springer Science+Business Media, 2009.
[25] F. M. Dekking, The spectrum of dynamical systems arising from substitu-tions of constant length, Zeitschrift f̈ur Wahrscheinlichkeitstheorie und Ver-wandte Gebiete, 41 (1978), pp. 221–239.
[26] V. Diekert, M. Volkov, andA. Voronkov, Computer Science - Theory and Applications, Theoretical Computer Science and General Issues, Springer-Verlag, Berlin Heidelberg, 1 ed., 2007.
[27] J. Dieudonné,Treatise on analysis. Vol. II, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Enlarged and corrected printing, Translated by I. G. Macdonald, With a loose erratum, Pure and Applied Mathematics, 10-II.
[28] F. Dreher, M. Kessebohmer̈ , A. Mosbach, T. Samuel,andM. Steffens, Reg-ularity of aperiodic minimal subshifts, Bulletin of Mathematical Sciences, (2017).
[29] F. Durand,Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Theory Dynam. Systems, 20 (2000), pp. 1061–
1078.
[30] M. Einsiedler and T. Ward, Ergodic theory with a view towards number theory, vol. 259 of Graduate Texts in Mathematics, Springer-Verlag London, Ltd., London, 2011.
[31] J. Elstrodt, Maß- und Integrationstheorie, vol. 7 of Springer-Lehrbuch, Springer-Verlag Berlin Heidelberg, 2011.
[32] K. Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, third ed., 2014. Mathematical foundations and applications.
[33] P. Glendinning, Topological conjugation of Lorenz maps by β-transformations, Math. Proc. Cambridge Philos. Soc., 107 (1990), pp. 401–413.