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Notation index
(a, b) (greatest common divisor),474
‖ · ‖ (distance to the nearest inte-ger), 365, 444
‖σ‖ (width), 11, 142, 507| · | (-adic absolute value), 420f g, 2g f , 2‖ · ‖1 (Manhattan norm), 2, 191‖ · ‖2 (Euclidean norm), 2‖ · ‖∞ (maximum norm), 2
1S (indicator function), 6, 455
AN (set of infinite words), 47Aσ,a (automaton associated with
the morphism σ), 142Aσ,a,τ (automaton associated
with the morphisms σ, τ),142
Adh(L) (adherence of L), 152Ap (canonical alphabet in base p),
37alph(u) (alphabet of u), 6alph(L) (alphabet of L), 14A≤n (words of length at most n),
4An (words of length n), 4A+ (free semigroup), 4A∗ (free monoid), 4AU (canonical alphabet), 110
b(n) (second difference of p(n)),173
Bad (set of badly approximablereal numbers), 444
BAL(σ, τ), 526B(x, R) (open ball), 2k (the signed digit −k), 41Bd (symmetrical digit alphabet
with largest digit d), 40Bk (language of the numeration
in base k), 109bq,i(w) (coefficients in the decom-
position of valS(w)), 120[w]x (cylinder), 29[u] (cylinder), 375[u]X (cylinder), 375BSn (u) (bispecial factors), 171BS′
n (u) (bispecial factors and ex-ceptional prefix), 172
χ[u ] (characteristic function), 375Cp(C × A) (the converter be-
tween C and A (in base p)),42
C(X, Z) (continuous maps), 352CYCLIC(σ) (set of cyclic letters),
510
d (distance on words), 7d−(w) (left valence), 171d+(w) (right valence), 171
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Notation index 595
δx (Dirac measure), 378, 474∂X (boundary), 2DG(X,T ) (dimension group),
352u⊕ v (digitwise addition), 42u! v (digitwise subtraction), 42Dp (set of all p-expansions of reals
in [0, 1)), 50
e(t) = e2πit , 476e (row vector in which all coordi-
nates equal 1), 512E−(w) (left extensions), 171, 231E+(w) (right extensions), 171E(w) (extension type), 172Ey (f1 , f2) (sandwich set), 188en (µ), 387ε (empty word), 4f ∼ g, 3Eu (self-similar tiling), 261EX (mean value of random vari-
able X), 483
Fq (finite field with q elements),453
(Fj )j≥0 (Fibonacci sequence),417
x (floor function), 2x (fractional part), 2fw (x) (frequency), 376, 380
Γc (self-replicating translationset), 263
Γe (self-similar translation set),261
[γ, i]∗ (tip), 263[γ, i]∗g (projected face), 263Gµ,f , 379Gn (Rauzy graph), 176Gn (X) (Rauzy graph), 384Gn (x) (Rauzy graph), 384GO (graph of overlaps), 300
GO(λ) (graph of overlaps), 301Gσ (prefix-suffix graph), 256
Hc (contracting space), 252He (expanding line), 252hσ (contraction), 252
∩ (intersection), 1[[i, j]] (interval of integers), 2Iσ (self-replicating multiple
tiling), 271
Kσn (x, y, z), 201
K≥a , 1K<a , 2K>a , 1K≤a , 2Km (a) (continuant), 429
Ln (u) (factors of length n), 164Lσ (x, y, z) (centric factors), 197L(u) (factors), 164Λr , 391〈σ1 , σ2〉 (monoid generated by
σ1 , σ2), 519Lb , 415L (set of pairs of real numbers
satisfying Littlewood’s con-jecture), 444
u < v (lexicographic order), 9u ≤p v (u is a prefix of v), 513u v (lexicographic order), 9u ≺ v (radix order), 9u v (radix order), 9L≤n (concatenation of at most n
words in L), 14L[n → nk ] (language s.t. U(n) =
nk ), 129L(A) (language recognised by A),
16L(X) (language), 374L(x) (language), 7
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596 Notation index
Ln (X) (words of L of length n),374
Ln (x) (factors of length n), 7log (logarithm), 3log2 (binary logarithm), 3Lp (set of all p-expansions), 39Lp
q(set of all p
q -expansions), 86Lq (language accepted from state
q), 117LSn (u) (left special factors), 171LS′
n (u) (left special factors andunioccurent prefix), 172
L∗ (Kleene star), 14Ln (power of a language), 14Lu (broken line), 253L(x) (language), 375
M(A) (adjacency matrix), 22Mσ (incidence matrix), 22, 191m(w) (bilateral multiplicity), 172E(X,T ) (ergodic invariant mea-
sures), 377Mf(s) (Mellin transform of f),
458Mβ (minimal polynomial), 62m (Lebesgue measure), 392M(X) (Borel measures), 30µk (Lebesgue measure), 252M(X,T ) (invariant measures),
377
Np(C) (the normaliser over thealphabet C (in base p)), 43
〈N〉p (p-expansion of N), 39〈N〉 p
q(pq -expansion of N), 86
νA,p (normalisation), 26
O(f), 2o(f), 3Ω(f), 2uω (concatenation), 8
ω(G) (infinite word generated byG), 506
ω(H), 508O(x) (orbit), 28, 374O(x) (orbit closure), 28
P (probability), 482P (abelianisation map), 6pu (n) (factor complexity), 164P(x) (Parikh vector), 191PER(w) (period of w), 516Φ(y) (normal distribution func-
tion), 487ϕ (Golden Ratio), 12π (permutation), 391πc (projection on the contracting
space), 252πe (projection on the expanding
line), 252πp (evaluation map), 37πp
q(evaluation map in the p
q nu-meration system), 86
P (n) (paths in Bratteli dia-grams), 358
u ∧ v (longest common prefix), 48x ∧ y (longest common prefix), 7Pσ (prefix-suffix edges), 256pX (n) (complexity function), 383px(n) (complexity function), 8
repk (representation in base k),109
repS (S-representation), 114, 117repq = repSq
, 118repU (U -representation), 109ρ(A) (spectral radius), 25, 512,
531ρ(Σ) (joint spectral radius), 533ρ(Σ) (joint spectral subradius),
534ρt , 533
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Notation index 597
ρt , 531ρt , 531ρ
t, 533
RSn (u) (right special factors),171
s(n) (first difference of p(n)), 173\ (set difference), 1S (shift map), 374,, (shuffle), 129σω (a), 11σω (b).σω (a), 11Sq = (Lq ,A,<), 118(X,S) (subshift), 374(Xx, S) (subshift generated by an
infinite word), 375S(X,T ) (states of the dimension
group), 362
T = R/Z (circle group), 453Θ(g), 2u (mirror image), 4, 446L (mirror image), 14Tλ,π (interval exchange map), 391Tσ (central tile), 253Tσ (i) (subtile), 253(T, S, λ) (overlap), 297[T, S, λ] (overlap equivalence
class), 298
U (lower unit cube), 274uβ (right eigenvector), 251UL (n) = U(n) (number of words
of length n in L), 118Un,ε , 393∪ (union), 1Uq (n) (number of words of length
n accepted from q), 117Uq ,r (n) (number of directed paths
of length n from q to r), 110
vp(n) (p-adic valuation), 474
valS (S-numerical value), 114valq = valSq
, 118vβ (left eigenvector), 251(V,E,≥) (ordered Bratteli dia-
gram), 327VL (n) = V(n) (number of words
of length at most n in L), 118Vq (n) (number of words of length
at most n accepted from q),117
Vσ (seed patch), 276vσ (Perron–Frobenius eigenvec-
tor), 24, 513VX (variance of random variable
X), 490
‖u‖ (weight of u), 47W(X,S) (weight functions), 379Wσ (two-piece seed patch), 282
XB (infinite path space associ-ated with an ordered Brattelidiagram B), 329
XmaxB , 329
XminB , 329
[x, i] (basic formal strand), 291ξa (real number whose b-ary ex-
pansion is given by the worda), 412
[x, i]g (basic geometric strand),253
Xσ (substitutive dynamical sys-tem), 32
Zβ ,d (the zero-automaton inbase β over the alpha-bet Bd), 62
Zp (the evaluator in base p), 40Z p
q(the evaluator in base p
q ), 89Zp,d (the zero-automaton in
base p over the alpha-bet Bd), 41
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598 Notation index
ζ(s) (Riemann zeta function), 459ζf ′ (Fibonacci continued frac-
tion), 411ζa (real number whose continued
fraction expansion is givenby the word a), 430
ζt′ (Thue–Morse continued frac-tion), 411
ζ(s, α) (Hurwitz zeta function),459
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General index
1-systemdefinition, 507Z-balanced, 519
a.e., see almost everywhereabelianisation map, 6abstract numeration system, 114accessible state, 16Adamczewski, B., 423additive function, 162, see q-
additiveadherence, 152adic, 399
dynamical system, 399Pascal, 404transformation, 324
adjacency matrix, 22Adjan, S. I., 33Aho–Corasick
algorithm, 550automaton, 549
Akiyama, S., 61, 83, 84, 249d’Alembert ratio test, 196algebraic
coincidence, 319conjugate, 23, 24integer, 24
Allouche, J.-P., 20, 161, 508, 512almost everywhere, 30alphabet, 3ancestor, 276
Angrand, P.-Y., 128ANS, see abstract numeration
systemapproximation algorithm
(k,l)-, 540non-existence of, 540
Arnoux, P., 8, 319atoms, 330automatic sequence, 19, 214, 450
q-automatic, 19, 138, 214, 452automatic word, see automatic
sequenceautomaton
Aho–Corasick, 549Buchi, 54complete, 17deterministic, 17deterministic with output, 19finite, 414local, 162trim, 16underlying input, 20
Avila, A., 366
badly approximable number, 444balanced pair
coincidence, 310algorithm, 310combinatorial, 310irreducible, 310one-letter, 310
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600 General index
Barabanov, N., 560Barat, G., 474Barbolosi, D., 463Barge, M., 258, 311, 319base, 37, 330
base-b expansion, 412odometer, 336
Bell, J., 161Berend, D., 63Berger, M. A., 538Bernoulli shift, 395Berry–Esseen inequality, 478Berstel, J., 8Berthe, V., 162, 527Bertrand, A., 59, 158Bertrand-Mathis, A., 70Bes, A., 73β-admissible, 57β-transformation, 56, 162Bezuglyi, S., 351bilateral multiplicity, 172binomial numeration system, 159Birkhoff ergodic theorem, 30, 377,
486Birkhoff, G., 377bispecial factor, 171
bound on their number, 216BK-property, 489, 491block growth, 163block triangular matrices, 536block-additive function, 472block-multiplicative function, 472Blondel, V. D., 539, 540Boasson, L., 152Borbely, T., 83, 84Borel measure, 596Borel, E., 413, 444, 449Boshernitzan, M., 335, 383, 384,
387bounded
gap, 7
letter, 206word, 198
Boyle, M., 354, 357, 358, 360Bratteli compactum, 329Bratteli diagram, 325
equivalence relation, 327incidence matrix, 326infinite path, 329isomorphic, 326morphism, 328ordered, 327properly ordered, 329range map, 326simple, 327source map, 326stationary, 337substitution, 337telescoping, 326, 328
Bratteli, O., 324, 325Bratteli–Vershik
BV, 329dynamical system, 329, 399
Bressaud, X., 369, 370, 404broken line, 253Brunotte, H., 83, 84Bruyere, V., 116, 153, 160Buchi automaton, 54Bugeaud, Y., 423, 449
canonical alphabet, 37canonical numeration system, 78Cantor
dynamical system, 325, 329linearly recurrent, 347
set, 7, 449space, 325, 329version of interval exchange,
350capacity of codes, 546Carroll, C. R., 335Carton, O., 129, 148
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General index 601
Cassaigne, J., 403central limit theorem, 487, 489,
494central tile, 253centric factor, 197Chacon
morphism, 24substitution, 24, 340, 342word, 342
Chacon, R. V., 342Chaika, J., 402Champernowne word, 148, 165,
395, 413Champernowne, D. G., 413characteristic word, 5, 221Charlier, E., 137, 150, 161Chebyshev norm, 188Cheung, Y., 397Choffrut, Ch., 72, 103, 159Chomsky hierarchy, 109Chomsky, N., 96Christol, G., 450circular shift, 170clopen
partition, 330set, 29, 325
CNS, see canonical numerationsystem
co-accessible state, 16co-sequential, 102Cobham’s conjecture, 414Cobham’s theorem, 181Cobham, A., 20, 27, 109, 138,
181, 215, 414, 508, 526coboundary, 352coboundary condition, 396code, 5, 183, 338
capacity, 546cirular, 338constrained, 546prefix, 5
coded subshift, 54coding, 10cofinal, 329coincidence
algebraic, 319combinatorial strong coinci-
dence, 258geometric, 319geometric strong coincidence,
293half-coincidence overlap, 298modular, 319overlap, 298strong, 318strong overlap, 300super, 319
combinatorial strong coincidencecondition, 258
common reducibility, 536compactum, 329comparable edges, 327complete automaton, 17complexity function, 118, 163,
164, 383action of a letter-to-letter mor-
phism, 182action of a non-erasing mor-
phism, 183action of an injective mor-
phism, 183computation, 221exponential, 169linear, 219maximal, 165non-decreasing, 165of a language, 166of a morphic word, 185, 209of a periodic word, 164, 239of a purely morphic word, 185,
209of a sparse word, 179
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602 General index
of an automatic word, 214of an interval translation map,
227of the Fibonacci word, 222of the Thue–Morse word, 224subadditivity, 168
complexity of a real number, 413concatenation, 4cone
positive, 351conjecture
finiteness property for the ca-pacity, 562
Pisot, 272conjugacy map, 31conjugate
algebraic, 23words, 516
consecutive, 358constant of expansivity, 345constrained codes, 546context-free language, 89continuant, 429continued fractions, 427convergent, 428converter, 41convex combination method, 555Coquet, J., 473Cornfeld, I. P., 349Cortez, M. I., 369, 370counting
function, 118, 129Culik II, K., 505, 515cycle, 278cyclic
letter, 510morphism, 510
cylinder, 29, 329, 375
D0Lω-equivalence problem, 506
language, 13ω-equivalent systems, 506prefix problem, 509system, 13, 149, 505
nearly primitive, 507prolongable, 506
Damanik, D., 343Dartnell, P., 348Daubechies, I., 532Davenport, H., 438De Bruijn graph, 547decimation, 123, 161deconnectable, 384Dekking, F. M., 318, 508Delange, H., 457Delone set, 260density, 470
logarithmic, 471de Bruijn, N. G., 384DFA, 17DFAO, 19Diamond, B., 258digit-conversion transducer, 41digitwise
addition, 42subtraction, 42
dimension group, 351, 352Diophantine approximation
uniform, 437directive language, 144Dirichlet series, 465Dirichlet’s theorem, 437Dirichlet, P. G. L., 436discrete hyperplane, 262distance, 7
ultrametric, 7division algorithm, 38dominant root condition, 67Downarowicz, T., 347, 349, 366Drmota, M., 493, 496, 500, 503Duchene, E., 161
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General index 603
Dumont–Thomas numeration,254, 319, 502, 513
Durand, F., 149, 335, 337, 338,344, 345, 348, 369, 370
dynamical systemBratteli–Vershik, 329Cantor, 325, 329conjugacy, 30induced, 325measure-theoretic, 30, 376stability, 537symbolic, 28, 53, 374topological, 377
equicontinuous, 347topological isomorphism, 30
edge, 325maximal, 329minimal, 329
Ehrenfeucht, A., 163, 201, 209,509, 511
Ehrenfeucht, Lee and Rozen-berg’s theorem, 209
Ei, H., 527eigenfunction
L2(µ), 364continuous, 364
eigenvalueL2(µ), 364continuous, 364Perron–Frobenius, 23
Eisiedler, M., 445elementary morphism, 509endomorphism, 10, see substitu-
tioneverywhere-growing, 199exponentially diverging, 200polynomially diverging, 200quasi-uniform, 199
entropy, 358topological, 60, 169
episturmian, 407equicontinuous, 347equivalent norms, 187equivalent orbit, 354erasing morphism, 10ergodic, 30, 377
Birkhoff theorem, 30, 377, 486individual ergodic theorem, 30,
377, 486theorem, 30, 377, 486
Euler–Lagrange’s theorem, 430evaluation map, 26, 37eventually periodic word, 8, 164Evertse, J.-H., 449everywhere-growing endomor-
phism, 199exceptional prefix, 172exduction, 397expansion
p-, 26, 39base-b, 412pq -, 86
expansive dynamical system, 345,399
expansive morphism, 199exponential language, 136exponentially diverging endomor-
phism, 200exponentially growing word, 194extension, 171extension type, 172extraction, 381extremal norm theorem, 560extremal number, 439
factor, 6, 163, 325centric, 197complexity, 163map, 325special, 171
factor graph, 176
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604 General index
factorial language, 15Fekete’s lemma, 169, 532Fekete, M., 532Ferenczi, S., 342, 365, 366, 420fiber, 148Fibonacci
continued fraction, 430sequence, 417word, 12, 168, 222, 223, 388,
410final state, 16finite automaton, 17, 414finite difference, 173finiteness property
for capacity, 562for joint spectral radius, 550geometric, 275property (F), 60, 321weak (W), 321
first entrance time map, 325first finite difference, 173Fischler, S., 443fixed point, 10folded β-expansion, 77Fomin, S. V., 349Forni, G., 366Forrest, A. H., 336fractional part, 49Fraenkel, A. S., 27, 159, 161frequency, 9Fretloh, D., 319Frid, A., 404Frougny, Ch., 27, 64, 71, 73, 75,
77, 111, 116, 128, 153, 160full shift, 28, 395function
additive, 162block-additive, 472block-multiplicative, 472co-sequential, 102completely
q-additive, 454q-multiplicative, 454
complexity, 118counting, 118generating, 96q-additive, 453q-automatic, 452q-multiplicative, 454q-regular, 453rational, 96, 508sequential, 102
fundamental lemma, 482Furstenberg, H., 396
Gambaudo, J.-M., 370gap, 7Gauss Lemma, 38Gelfand, I. M., 187Gelfond, A. O., 499genealogical order, 9generalised spectral radius, 532generalised spectral subradius,
534generating function, 96generation of bispecial factors,
236generic, 380geometric strong coincidence, 293GIFS, 255
substitution, 265Giordano, T., 351, 354Gjerde, R., 349, 350Glasner, E., 354Golden Ratio, 12, 436Goldwurm, M., 159Grabner, P. J., 162, 463, 474graph
overlaps, 300prefix-suffix, 256two-piece ancestor, 282
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General index 605
graph-directed iterated functionsystem, 255
greedyβ-expansion, 56algorithm, 38, 49
Grillenberger, C., 169, 391group ordered, 351growing word, 198
Haar measure, 486Hahn, F., 391half-coincidence overlap, 298Hamming weight, 47Handelman, D., 357, 358, 360Hansel, G., 27, 116, 148, 149, 153,
160Harju, T., 161, 505, 515, 529Hausdorff dimension, 445HD0L
language, 13sequence, 507system, 149, 507
prolongable, 507Hedlund, G. A., 163, 348height, 253Heinis, A., 220Herman, R. H., 324, 330, 333, 335Hollander, M., 68, 69, 111, 272,
321Holton, C., 335, 366homomorphism of monoids, 4Honkala, J., 160, 505, 508, 526,
529Horner scheme, 39Host, B., 337, 338, 344, 345, 364,
365, 369hyperplane
discrete, 262stepped, 262
i.d.o.c. property, 392
immortal letter, 246incidence matrix, 22, 191, 200
of a Bratteli diagram, 326independence condition, 396indicator function, 6, 356individual ergodic theorem, 30,
377, 486induced
dynamical system, 325map, 325, 398
infinite wordautomatic, 214q-automatic, 214Champernowne, 165D0L, 506Fibonacci, 168, 222, 223HD0L, 508left-infinite, 180lexicographically shift maxi-
mal, 54lsm-word, 54morphic, 181paperfolding, 182period-doubling, 244periodic, 164purely morphic, 181recurrent, 177, 186sparse, 179Sturmian, 167, 177, 223Thue–Morse, 224
inflation factor, 23injective morphism, 183integer
multiplicativelydependent, 27independent, 27
representation, 86integral part, 49internal alphabet, 214interval exchange
Cantor version, 350
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606 General index
map, 391transformation, 349
invarianceunder similarity, 535
invariantmeasure, 30subset, 377
irreduciblematrix, 23morphism, 23permutation, 392set of matrices, 536substitution, 23
isomorphism, 325measure-theoretic, 31topological, 30, 325
iterated function systemgraph-directed, 255
Ito, S., 319, 527Iwanik, A., 366
Jacobs, K., 349Jewett, R., 381Johansen, O., 349, 350joint spectral radius, 533
capacity, 548computation, 552finiteness property, 550introduction, 531partition function, 541repetition-free words, 543undecidability, 540
joint spectral subradiusdefinition, 533, 534
Jordan normal form, 187Jungers, R., 538
k-kernel, see q-kernelKabore, I., 403Kakutani equivalent, 335Kakutani, S., 330
Kakutani–Rokhlin partition, 401Karki, T., 150Katai, I., 488Katok, A., 392, 445Katznelson, Y., 391Keane, M., 349, 393kernel
k-kernel, 147q-kernel, 452S-kernel, 148
Khintchine, A. Ya., 430Kleene star, 14Kleene, S., 18Kozyakin, V. S., 539, 560Krieger, D., 161Krieger, W., 381Kronecker’s theorem, 264Kubilius model, 481Kubilius, J., 438Kwapisz, J., 311, 319Kwiatkowski, J., 351Kurka, P., 345, 348
Lacroix, L., 366Lagarias, J. C., 532, 551Lang, S., 361language, 13, 341, 374
adherence, 152context-free, 89D0L, 13directive, 144exponential, 136extendable, 374factorial, 15, 374finite, 14HD0L, 13infinite, 14of an infinite word, 164polynomial, 136prefix-closed, 15ray, 96
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General index 607
slender, 15, 134, 160, 525sparse, 136suffix-closed, 15with bounded growth, 15, 96
Larcher, G., 464Lecomte, P., 137, 152Lee, J.-Y., 319Lee, K. P., 163, 201, 209left
extension, 171special factor, 171valence, 171
left-infinite word, 180length, 4Lenz, D., 343Leroux, J., 161Lesigne, E., 389letter, 3
cyclic, 510letter-to-letter
morphism, 10, 182transducer, 20
level, 330Levy metric, 488lexicographic
map, 329order, 9, 48, 186, 328
Liardet, P., 162Lindenmayer systems, 149Lindenmayer, A., 505Lindenstauss, E., 445linearly recurrent
Cantor dynamical system, 347Linna, M., 161, 509, 529Liouville’s inequality, 415Liouville, J., 415Littlewood’s conjecture, 444Livshits, A. N., 309, 340local automaton, 162locally finite, 259logarithmic density, 471
logarithmically syndetic set, 227looping morphism, 517Loraud, N., 67, 111Lothaire, M., 3, 349, 509Luca, F., 423
Maass, A., 347, 348, 369, 370Maes, A., 137, 148, 151Mahler, K., 423, 449Manhattan norm, 2, 191map
abelianisation, 6factor, 325first entrance time, 325induced, 325lexicographic, 329range, 326source, 326Vershik, 329
Markov compacta, 324Masur, H., 393matrix
adjacency, 22incidence, 22irreducible, 23primitive, 23
Mauduit, Ch., 148, 161, 365, 420measure invariant, 30measure-theoretic
dynamical system, 30factor, 31isomorphism, 31
Medynets, K., 350, 351Mellin–Perron summation for-
mula, 466Mendes France, M., 502Michaux, C., 160Miller, A., 161Miller, G. A., 96minimal, 375
dynamical system, 29
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608 General index
word, 7mirror, 4, 446
formula, 429Modified Division algorithm, 85monoid, 4
free, 4morphism, 4of matrices, 531
Monteil, T., 383, 384Moody, R.V., 319morphic word, 11, 181morphism, 10, see also endomor-
phismcyclic, 510elementary, 509erasing, 10expansive, 199fixed point, 10growing, 517injective, 183invertible, 527irreducible, 23letter-to-letter, 10, 182loop-free, 517looping, 517non-erasing, 10non-trivial, 511of ordered groups with order
unit, 351Pisot, 24primitive, 23prolongable, 10, 11proper, 338read on a Bratteli diagram, 328simplifiable, 509uniform, 10unit, 25
Morse and Hedlund theorem, 166,175, 199
Morse substitution, 340Morse, M., 163, 224, 340, 348
mortal letter, 246Mosse, B., 365multi-graph, 110multi-scale quasiperiodic, 408multiple tiling, 259multiplicative function, see q-
multiplicativemultiplicatively
dependent integers, 27independent integers, 27
multiplicityof a bispecial factor, 172
Mela, X., 405
natural coding, 392neutral bispecial factor, 173Nicolay, S., 161Nivat, M., 152Nogueira, A., 365, 366non-algebraicity, 539non-defective, 560non-erasing morphism, 10non-periodic word, 8non-transient letter, 246non-trivial morphism, 511norm, 186
submultiplicative, 25, 531normal
matrices, 554number, 413
normalisation, 26, 71, 112, 160normaliser, 43Novikov, P. S., 33Nowakowski, R. J., 161number
Parry, 58Perron, 104Pisot, 24, 35, 115Salem, 104triangular, 117
numeration
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General index 609
scale, 65numeration system
p-ary, 26abstract, 114adic, 115canonical (CNS), 78Dumont–Thomas, 319, 502linear, 67positional, 110scale, 65, 110
numerical value, 26S-numerical value, 114
occurrence, 341odometer, 162, 336, 401, 486, 501
base, 336ω-equivalent
D0L systems, 506HDOL systems, 149
one-sided shift, 28orbit, 374
equivalent, 354strongly, 354
of a word, 28order
genealogical, 9lexicographic, 9, 328radix, 9unit, 351, 352
ordered, 327Bratteli diagram, 327group, 351properly, 329
orderingconsecutive, 358
ordinary bispecial factor, 173, 234Oseledec, V. I., 391overlap, 297, 419, 543
graph, 300coincidence, 298equivalent, 298
half-coincidence, 298Oxtoby, J., 381
palindrome, 4, 434palindromic density, 443Pansiot’s theorem, 185Pansiot, J.-J., 161, 185, 508, 509,
529paperfolding word, 182Parikh mapping, 6Parikh vector, 191Parry number, 58Parry, W., 58partial quotients, 428partition
clopen, 330Kakutani–Rohlin, 330KR, 330
patch, 260ancestor, 281equivalent, 260
path, 16Bratteli diagram, 329cofinal, 329label, 16space, 329successful, 16
Paun, G., 134Pell equation, 137period, 8period cycle, 164period-doubling word, 244periodic word, 8, 164Perron number, 104Perron–Frobenius
eigenvalue, 23normalised eigenvector, 24theorem, 23, 250
Petersen, K., 405Petho, A., 83, 84p-expansion, 26, 39
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610 General indexpq -expansion, 86Pisot
conjecture, 272morphism, 24number, 24, 35, 115
unit, 24substitution, 24Vijayaraghavan number, 24
polynomial language, 136polynomially bounded word, 194polynomially diverging endomor-
phism, 200positional numeration system,
110positive cone, 351positive uniform frequencies, 390power of a set, 535powers of two, 221prefix, 6
-closed language, 15exceptional, 172proper, 6unioccurrent, 171
prefix-suffix graph, 256preperiod, 8, 164primitive
matrix, 23morphism, 23substitution, 23word, 164, 178
problemD0L ω-equivalence, 506D0L prefix, 509HD0L ω-equivalence, 149
prolongableD0L system, 506morphism, 10
propersubstitution, 338
properlyordered, 329
property (F), see finiteness prop-erty
pumping lemma, 18, 89purely
morphic word, 11, 181substitutive word, 11, 505
Putnam, I. F., 324, 330, 333, 335,351, 354
Pytheas Fogg, N., 509
q-additive function, 453completely, 454
q-automatic function, 452q-automatic sequence, 452q-kernel, 452q-multiplicative function, 454
completely, 454q-regular function/sequence, 453quasi-greedy expansion, 58quasi-recurrent word, 506quasi-uniform endomorphism,
199Queffelec, M., 309, 434
Rabin, M. O., 17radix
order, 9, 39point, 48
Rampersad, N., 161range map, 326rank
topological, 347infinite, 347
ranking, 159Rao, H., 319, 527rational function, 96, 508Ratner, M., 406Rauzy
fractal, 253graph, 176, 384
Rauzy, G., 176, 248, 338, 384
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General index 611
Ravikumar, B., 161recurrence
linear, 408uniform, 375
recurrent word, 7, 171, 177, 186reducible
set of matrices, 536redundancy transducer, 44regular function/sequence, see q-
regularregular language, 88relatively dense, 260repetition, 419, 543repetitive, 260representation
U -, 65, 109S-, 114, 117p-ary, 26, see expansionBV, 331greedy, 26integer, 86
return word, 178, 341reversal, 4Ridout’s Theorem, 420Ridout, D., 420right
extension, 171special factor, 171valence, 171
right transducer, 21right context, 88Rigo, M., 137, 148, 150, 158, 317,
526Rohlin, V. A., 330Rosenthal, A., 381Rota, G. C., 531, 536Roth’s Theorem, 416Roth, K. F., 416Roy, D., 438Rozenberg, G., 163, 201, 209, 509,
511
Rudin–Shapiro word, 388Rudolph, D. J., 366, 406Ruohonen, K., 508
s-adic construction, 226S-recognisable set, 117Seebold, P., 8Sadun, L., 321Sakarovitch, J., 77, 128Salem number, 104Salomaa and Soittola’s theorem,
191Salomaa, A., 134, 191, 509Salon, O., 150sandwich set, 180sandwich set theorem, 188Sataev, E., 397scale, 65, 110scaling property, 535Schutzenberger, M.-P., 33Schlickewei, H. P., 450Schmidt’s subspace theorem, 422Schmidt, K., 59, 158Schmidt, W. M., 434, 438Scott, D., 17second finite difference, 173seed patch, 276, 282self-replicating multiple tiling,
271semigroup, 4
of matrices, 531sequence
2-regular, 545q-automatic, 19, 138, 452q-regular, 453S-automatic, 138
sequential, 102transducer, 20
setCantor, 7eventually periodic, 8
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612 General index
recognisable, 27, 109S-recognisable, 117U -recognisable, 73, 110
syndetic, 7Shallit, J., 20, 67, 111, 161, 508,
512shape of a Rauzy graph, 178shift, 28, 139, 374
circular, 170full, 28one-sided, 28two-sided, 28
shift radix system, 83shuffle, 129Siegel, A., 249simple
Bratteli diagram, 327hat, 327
simplifiable morphism, 509Sinai, Y. V., 349Sing, B., 319sink, 18Sirvent, V., 296Skau, C. F., 324, 330, 333, 335,
337, 338, 344, 345, 351, 354S-kernel, 148skew-product, 498slender language, 15, 134, 160,
525SOE, see strongly orbit equiva-
lentSoittola, M., 191Solomyak, B., 71, 77, 153, 272,
296, 319, 370source map, 326sparse language, 136sparse word, 179special factor, 171
bound on their number, 219spectral radius, 25, 187, 512square, 4
SRS, see shift radix systemstammering, 419state, 362
accessible, 16co-accessible, 16final, 16terminal, 16
stationary, 337Bratteli diagram, 337
Steiner, W., 137, 158stepped hyperplane, 262strand
basic formal, 291basic geometric, 253formal, 291geometric, 291
Strang, G., 531, 536strictly ergodic, 390strong bispecial factor, 172strong mixing, 408strongly orbit equivalent, 354Sturmian
expansion, 421subshift, 348word, 8, 167, 177, 223, 388
subadditive function, 168subadditivity, 532submultiplicative norm, 25, 531subshift, 28, 53, 374
aperiodic, 28, 337, 385coded, 54conjugate, 30entropy, 60finite type, 28, 53, 395periodic, 28, 337sofic, 28, 54, 395substitution, 343Toeplitz, 349
substitution, 10, see endomor-phism, see morphism
Chacon, 342
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General index 613
GIFS, 265invertible, 527irreducible, 23Morse, 340Pisot, 24
irreducible, 25reducible, 25
primitive, 23proper, 338read on a stationary Bratteli di-
agram, 337subshift, 343unit, 25
substitutiveword, 11word sequence, 507
subwordcomplexity, 163scattered, 6
successor, 128suffix, 6
-closed language, 15proper, 6
Sugisaki, F., 360sum-of-digits, 162, 452, 457, 467,
499, 502switched linear system, 538symbol, 3symbolic dynamical system, 28synchronisation lemma, 221, 229syndetic, 227
set, 7system
D0L, 13, 149, 505nearly primitive, 507prolongable, 506
HD0L, 149, 507
tag sequence, 138telescoping, 326, 328Tenenbaum, G., 457
terminal state, 16theorem
Cobham, 181Ehrenfeucht, Lee and Rozen-
berg, 209Grillenberger, 169Kronecker, 264Morse and Hedlund, 166, 175,
199Pansiot, 185Perron–Frobenius, 23, 250Salomaa and Soittola, 191sandwich set, 188
Theys, J., 534, 539Thomas, W., 129, 148Thue, A., 33, 224Thue–Morse
Bratteli diagram, 340continued fraction, 434word, 12, 224, 388, 410, 560
Thurston, W., 248Thuswaldner, J., 83, 84Tichy, R. F., 162tiles, 259tiling, 162, 259
synchronisation, 298lattice multiple, 320multiple, 259patch, 260property, 272self-replicating multiple, 271self-similar, 261
tip, 263Toeplitz
subshift, 349word, 170, 349
topologicalconjugacy, 30dynamical system, 29entropy, 60, 169, 390isomorphism, 30
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614 General index
total irrationality, 392tower, 330
level, 330Transcendence criterion, 423transducer, 20
co-sequential, 102digit-conversion, 41letter-to-letter, 20, 101right, 21sequential, 20, 102
transformationβ-, 56, 162adic, 324
transient part, 164transition
function, 17relation, 16structure, 141
triangularmatrices, 536
joint spectral radius of, 554number, 117
trie, 116trim, 16Tsitsiklis, J. N., 539, 540Turan–Kubilius inequality, 484two-piece ancestor graph, 282
U -recognisable, 73, 110U -representation, 65underlying input automaton, 20uniform Diophantine approxima-
tion, 437uniform frequencies, 376uniformly recurrent word, 7uniformly discrete, 260unioccurrent prefix, 171unique ergodicity, 30, 32, 380unit, 24
morphism, 25order, 351, 352
substitution, 25universal counter-example, 179
valence, 171Veech, W., 393, 396Vershik map, 329Vershik, A. M., 324, 329, 330,
340, 399vertex, 325Villemaire, R., 160
Walters, P., 357, 358Wang, Y., 538, 551weak (W), see finiteness propertyweak bispecial factor, 172weight, 47, 475weight function, 379
on a graph, 393Weiss, B., 354Wen, Z., 527width of a morphism, 11Wirsing, E., 451word, 3, see sequence
automatic, 19S-automatic, 138
β-admissible, 57bi-infinite, 5bounded, 198Chacon, 24, 342Champernowne, 148, 395, 413characteristic, 5comparable, 522concatenation, 4D0L infinite word, 506distance, 7empty, 4eventually periodic, 8exponentially growing, 194factor, 6Fibonacci, 12, 410growing, 198
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General index 615
infinitelexicographically shift maxi-
mal, 54lsm-word, 54one-sided, 5two-sided, 5
length, 4minimal, 7mirror, 4morphic, 11
purely, 11nearly periodic, 515non-periodic, 8ω-equivalent, 149period, 8periodic, 8polynomially bounded, 194prefix, 6preperiod, 8primitive, 164, 178purely substitutive, 11, 505quasi-recurrent, 506recurrent, 7reversal, 4Sturmian, 8substitutive, 11
purely, 11, 505subword (scattered), 6suffix, 6Thue–Morse, 12, 410, 560Toeplitz, 349uniformly recurrent, 7
Wythoff’s game, 161
Yu, S., 13
Zamboni, L., 335, 366Z-balanced 1-systems, 519zero automaton, 41, 62, 288zero spectral radius, 553
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