Combinatorics, Automata and Number Theoryassets.cambridge.org › 97805215 › 15979 › index ›...

22
Notation index (a, b) (greatest common divisor), 474 · (distance to the nearest inte- ger), 365, 444 σ (width), 11, 142, 507 |·| (-adic absolute value), 420 f g,2 g f ,2 · 1 (Manhattan norm), 2, 191 · 2 (Euclidean norm), 2 · (maximum norm), 2 1 S (indicator function), 6, 455 A N (set of infinite words), 47 A σ,a (automaton associated with the morphism σ), 142 A σ,a,τ (automaton associated with the morphisms σ, τ ), 142 Adh(L) (adherence of L), 152 A p (canonical alphabet in base p), 37 alph(u) (alphabet of u), 6 alph(L) (alphabet of L), 14 A n (words of length at most n), 4 A n (words of length n), 4 A + (free semigroup), 4 A (free monoid), 4 A U (canonical alphabet), 110 b(n) (second difference of p(n)), 173 Bad (set of badly approximable real numbers), 444 BAL(σ, τ ), 526 B(x,R) (open ball), 2 k (the signed digit k), 41 B d (symmetrical digit alphabet with largest digit d), 40 B k (language of the numeration in base k), 109 b q,i (w) (coefficients in the decom- position of val S (w)), 120 [w] x (cylinder), 29 [u] (cylinder), 375 [u] X (cylinder), 375 BS n (u) (bispecial factors), 171 BS n (u) (bispecial factors and ex- ceptional prefix), 172 χ [ u ] (characteristic function), 375 C p (C × A) (the converter be- tween C and A (in base p)), 42 C(X, Z) (continuous maps), 352 CYCLIC(σ) (set of cyclic letters), 510 d (distance on words), 7 d (w) (left valence), 171 d + (w) (right valence), 171 www.cambridge.org © in this web service Cambridge University Press Cambridge University Press 978-0-521-51597-9 - Combinatorics, Automata and Number Theory Edited by Valérie Berthé and Michel Rigo Index More information

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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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