Combinatorics, Automata and Number Theory Edited by Valérie Berthé and Michel Rigo Index More Information

Cambridge University Press 978-0-521-51597-9 - Combinatorics, Automata and Number Theory Edited by Valérie Berthé and Michel Rigo Index More information

Notation index

(a, b) (greatest common divisor), b(n) (second diﬀerence of p(n)), 474 173 ·(distance to the nearest inte- Bad (set of badly approximable ger), 365, 444 real numbers), 444 σ (width), 11, 142, 507 BAL(σ, τ ), 526

|·| (-adic absolute value), 420 B(x,R) (open ball), 2 f g,2 k (the signed digit −k), 41

g f,2 Bd (symmetrical digit alphabet · 1 (Manhattan norm), 2, 191 with largest digit d), 40 · 2 (Euclidean norm), 2 Bk (language of the numeration ·∞ (maximum norm), 2 in base k), 109 bq,i(w) (coeﬃcients in the decom- 1S (indicator function), 6, 455 position of valS (w)), 120

[w]x (cylinder), 29 N A (set of inﬁnite words), 47 [u] (cylinder), 375 A σ,a (automaton associated with [u]X (cylinder), 375 the morphism σ), 142 BSn (u) (bispecial factors), 171 A (automaton associated σ,a,τ BSn (u) (bispecial factors and ex- with the morphisms σ, τ ), ceptional preﬁx), 172 142

Adh(L) (adherence of L), 152 χ[u] (characteristic function), 375 Ap (canonical alphabet in base p), Cp (C × A) (the converter be- 37 tween C and A (in base p)), alph(u) (alphabet of u), 6 42 alph(L) (alphabet of L), 14 C(X, Z) (continuous maps), 352 ≤ A n (words of length at most n), CYCLIC(σ) (set of cyclic letters), 4 510 An (words of length n), 4 A+ (free semigroup), 4 d (distance on words), 7 A∗ (free monoid), 4 d−(w) (left valence), 171 + AU (canonical alphabet), 110 d (w) (right valence), 171

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-51597-9 - Combinatorics, Automata and Number Theory Edited by Valérie Berthé and Michel Rigo Index More information

Notation index 595

δx (Dirac measure), 378, 474 GO(λ) (graph of overlaps), 301 ∂X (boundary), 2 Gσ (preﬁx-suﬃx graph), 256 DG(X, T) (dimension group), H 352 c (contracting space), 252 H u ⊕ v (digitwise addition), 42 e (expanding line), 252 u ! v (digitwise subtraction), 42 hσ (contraction), 252 D (set of all p-expansions of reals p ∩ (intersection), 1 in [0, 1)), 50 [[ i, j]] (interval of integers), 2 I e(t)=e2πit, 476 σ (self-replicating multiple e (row vector in which all coordi- tiling), 271

nates equal 1), 512 σ − K (x, y, z), 201 E (w) (left extensions), 171, 231 n K≥ ,1 E+ (w) (right extensions), 171 a K ,2 E(w) (extension type), 172 a,1 Ey (f1 ,f2 ) (sandwich set), 188 K≤a ,2 en (µ), 387 K (a) (continuant), 429 ε (empty word), 4 m ∼ f g,3 L (u) (factors of length n), 164 E n u (self-similar tiling), 261 Lσ (x, y, z) (centric factors), 197 E X (mean value of random vari- L(u) (factors), 164 able X), 483 Λr , 391 σ1 ,σ2 (monoid generated by Fq (finite field with q elements), 453 σ1 ,σ2 ), 519 Lb , 415 (Fj )j≥0 (Fibonacci sequence), 417 L (set of pairs of real numbers x (floor function), 2 satisfying Littlewood’s con- {x} (fractional part), 2 jecture), 444 u

596 Notation index

Ln (X)(wordsofL of length n), ω(G) (inﬁnite word generated by 374 G), 506

Ln (x) (factors of length n), 7 ω(H), 508 log (logarithm), 3 O(x) (orbit), 28, 374 O log2 (binary logarithm), 3 (x) (orbit closure), 28 Lp (set of all p-expansions), 39 p P L p (set of all -expansions), 86 (probability), 482 q q Lq (language accepted from state P (abelianisation map), 6 q), 117 pu (n) (factor complexity), 164

LSn (u) (left special factors), 171 P(x) (Parikh vector), 191 LSn (u) (left special factors and PER(w)(periodofw), 516 unioccurent preﬁx), 172 Φ(y) (normal distribution func- L∗ (Kleene star), 14 tion), 487 Ln (power of a language), 14 ϕ (Golden Ratio), 12

Lu (broken line), 253 π (permutation), 391 L(x) (language), 375 πc (projection on the contracting space), 252

M(A) (adjacency matrix), 22 πe (projection on the expanding Mσ (incidence matrix), 22, 191 line), 252 m(w) (bilateral multiplicity), 172 πp (evaluation map), 37 p E(X, T) (ergodic invariant mea- π p (evaluation map in the nu- q q sures), 377 meration system), 86 Mf(s) (Mellin transform of f), P (n) (paths in Bratteli dia- 458 grams), 358

Mβ (minimal polynomial), 62 u ∧ v (longest common prefix), 48 m (Lebesgue measure), 392 x ∧ y (longest common prefix), 7 M(X) (Borel measures), 30 Pσ (prefix-suffix edges), 256 µk (Lebesgue measure), 252 pX (n) (complexity function), 383 M(X, T) (invariant measures), px (n) (complexity function), 8 377 repk (representation in base k), Np (C) (the normaliser over the 109 alphabet C (in base p)), 43 repS (S-representation), 114, 117 N p (p-expansion of N), 39 repq =repS , 118 p q N p ( -expansion of N), 86 repU (U-representation), 109 q q νA,p (normalisation), 26 ρ(A) (spectral radius), 25, 512, 531 O(f), 2 ρ(Σ) (joint spectral radius), 533 o(f), 3 ρˇ(Σ) (joint spectral subradius), Ω(f), 2 534 ω u (concatenation), 8 ρˇt , 533

Notation index 597

ρˆt , 531 valS (S-numerical value), 114

ρt , 531 valq =valSq , 118 ρ , 533 v (left eigenvector), 251 t β RSn (u) (right special factors), (V,E,≥) (ordered Bratteli dia- 171 gram), 327 VL (n)=V(n) (number of words s(n) (first difference of p(n)), 173 of length at most n in L), 118 \ (set difference), 1 Vq (n) (number of words of length S (shift map), 374 at most n accepted from q), ,, (shuffle), 129 117 σω (a), 11 Vσ (seed patch), 276 σω (b).σω (a), 11 vσ (Perron–Frobenius eigenvec- S q =(Lq ,A,<), 118 tor), 24, 513 (X, S) (subshift), 374 VX (variance of random variable (Xx ,S) (subshift generated by an X), 490 infinite word), 375 S(X, T) (states of the dimension u (weight of u), 47 group), 362 W(X, S) (weight functions), 379 Wσ (two-piece seed patch), 282 T = R/Z (circle group), 453 Θ(g), 2 XB (infinite path space associ- u˜ (mirror image), 4, 446 ated with an ordered Bratteli L˜ (mirror image), 14 diagram B), 329 max Tλ,π (interval exchange map), 391 XB , 329 min Tσ (central tile), 253 XB , 329 Tσ (i) (subtile), 253 [x,i] (basic formal strand), 291 (T,S,λ) (overlap), 297 ξa (real number whose b-ary ex- [T,S,λ] (overlap equivalence pansion is given by the word class), 298 a), 412 [x,i]g (basic geometric strand), U (lower unit cube), 274 253 uβ (right eigenvector), 251 Xσ (substitutive dynamical sys- UL (n)=U(n) (number of words tem), 32 of length n in L), 118 Z (the zero-automaton in Un,ε, 393 β,d ∪ (union), 1 base β over the alphabet B ), 62 Uq (n) (number of words of length d Zp (the evaluator in base p), 40 n accepted from q), 117 p U Z p (the evaluator in base ), 89 q,r(n) (number of directed paths q q Z of length n from q to r), 110 p,d (the zero-automaton in base p over the alpha- vp (n)(p-adic valuation), 474 bet Bd ), 41

598 Notation index

ζ(s) (Riemann zeta function), 459 ζf (Fibonacci continued fraction), 411 ζa (real number whose continued fraction expansion is given by the word a), 430 ζt (Thue–Morse continued fraction), 411 ζ(s, α) (Hurwitz zeta function), 459

General index

1-system Angrand, P.-Y., 128 definition, 507 ANS, see abstract numeration Z-balanced, 519 system approximation algorithm a.e., see almost everywhere (k,l)-, 540 abelianisation map, 6 non-existence of, 540 abstract numeration system, 114 Arnoux, P., 8, 319 accessible state, 16 atoms, 330 Adamczewski, B., 423 automatic sequence, 19, 214, 450 additive function, 162, see q- q-automatic, 19, 138, 214, 452 additive automatic word, see automatic adherence, 152 sequence adic, 399 automaton dynamical system, 399 Aho–Corasick, 549 Pascal, 404 Büchi, 54 transformation, 324 complete, 17 adjacency matrix, 22 deterministic, 17 Adjan, S. I., 33 deterministic with output, 19 Aho–Corasick finite, 414 algorithm, 550 local, 162 automaton, 549 trim, 16 Akiyama, S., 61, 83, 84, 249 underlying input, 20 d’Alembert ratio test, 196 Avila, A., 366 algebraic coincidence, 319 badly approximable number, 444 conjugate, 23, 24 balanced pair integer, 24 coincidence, 310 Allouche, J.-P., 20, 161, 508, 512 algorithm, 310 almost everywhere, 30 combinatorial, 310 alphabet, 3 irreducible, 310 ancestor, 276 one-letter, 310

600 General index

Barabanov, N., 560 letter, 206 Barat, G., 474 word, 198 Barbolosi, D., 463 Boyle, M., 354, 357, 358, 360 Barge, M., 258, 311, 319 Bratteli compactum, 329 base, 37, 330 Bratteli diagram, 325 base-b expansion, 412 equivalence relation, 327 odometer, 336 incidence matrix, 326 Bell, J., 161 infinite path, 329 Berend, D., 63 isomorphic, 326 Berger, M. A., 538 morphism, 328 Bernoulli shift, 395 ordered, 327 Berry–Esseen inequality, 478 properly ordered, 329 Berstel, J., 8 range map, 326 Berthé, V., 162, 527 simple, 327 Bertrand, A., 59, 158 source map, 326 Bertrand-Mathis, A., 70 stationary, 337 Bès, A., 73 substitution, 337 β-admissible, 57 telescoping, 326, 328 β-transformation, 56, 162 Bratteli, O., 324, 325 Bezuglyi, S., 351 Bratteli–Vershik bilateral multiplicity, 172 BV, 329 binomial numeration system, 159 dynamical system, 329, 399 Birkhoff ergodic theorem, 30, 377, Bressaud, X., 369, 370, 404 486 broken line, 253 Birkhoff, G., 377 Brunotte, H., 83, 84 bispecial factor, 171 Bruyère, V., 116, 153, 160 bound on their number, 216 Büchi automaton, 54 BK-property, 489, 491 Bugeaud, Y., 423, 449 block growth, 163 block triangular matrices, 536 canonical alphabet, 37 block-additive function, 472 canonical numeration system, 78 block-multiplicative function, 472 Cantor Blondel, V. D., 539, 540 dynamical system, 325, 329 Boasson, L., 152 linearly recurrent, 347 Borbély, T., 83, 84 set, 7, 449 Borel measure, 596 space, 325, 329 Borel, E.,´ 413, 444, 449 version of interval exchange, Boshernitzan, M., 335, 383, 384, 350 387 capacity of codes, 546 bounded Carroll, C. R., 335 gap, 7 Carton, O., 129, 148

General index 601

Cassaigne, J., 403 coded subshift, 54 central limit theorem, 487, 489, coding, 10 494 cofinal, 329 central tile, 253 coincidence centric factor, 197 algebraic, 319 Chacon combinatorial strong coinci- morphism, 24 dence, 258 substitution, 24, 340, 342 geometric, 319 word, 342 geometric strong coincidence, Chacon, R. V., 342 293 Chaika, J., 402 half-coincidence overlap, 298 Champernowne word, 148, 165, modular, 319 395, 413 overlap, 298 Champernowne, D. G., 413 strong, 318 characteristic word, 5, 221 strong overlap, 300 Charlier, E., 137, 150, 161 super, 319 Chebyshev norm, 188 combinatorial strong coincidence Cheung, Y., 397 condition, 258 Choffrut, Ch., 72, 103, 159 common reducibility, 536 Chomsky hierarchy, 109 compactum, 329 Chomsky, N., 96 comparable edges, 327 Christol, G., 450 complete automaton, 17 circular shift, 170 complexity function, 118, 163, clopen 164, 383 partition, 330 action of a letter-to-letter mor- set, 29, 325 phism, 182 CNS, see canonical numeration action of a non-erasing mor- system phism, 183 co-accessible state, 16 action of an injective mor- co-sequential, 102 phism, 183 Cobham’s conjecture, 414 computation, 221 Cobham’s theorem, 181 exponential, 169 Cobham, A., 20, 27, 109, 138, linear, 219 181, 215, 414, 508, 526 maximal, 165 coboundary, 352 non-decreasing, 165 coboundary condition, 396 of a language, 166 code, 5, 183, 338 of a morphic word, 185, 209 capacity, 546 of a periodic word, 164, 239 cirular, 338 of a purely morphic word, 185, constrained, 546 209 prefix, 5 of a sparse word, 179

602 General index

of an automatic word, 214 language, 13 of an interval translation map, ω-equivalent systems, 506 227 prefix problem, 509 of the Fibonacci word, 222 system, 13, 149, 505 of the Thue–Morse word, 224 nearly primitive, 507 subadditivity, 168 prolongable, 506 complexity of a real number, 413 Damanik, D., 343 concatenation, 4 Dartnell, P., 348 cone Daubechies, I., 532 positive, 351 Davenport, H., 438 conjecture De Bruijn graph, 547 finiteness property for the ca- decimation, 123, 161 pacity, 562 deconnectable, 384 Pisot, 272 Dekking, F. M., 318, 508 conjugacy map, 31 Delange, H., 457 conjugate Delone set, 260 algebraic, 23 density, 470 words, 516 logarithmic, 471 consecutive, 358 de Bruijn, N. G., 384 constant of expansivity, 345 DFA, 17 constrained codes, 546 DFAO, 19 context-free language, 89 Diamond, B., 258 continuant, 429 digit-conversion transducer, 41 continued fractions, 427 digitwise convergent, 428 addition, 42 converter, 41 subtraction, 42 convex combination method, 555 dimension group, 351, 352 Coquet, J., 473 Diophantine approximation Cornfeld, I. P., 349 uniform, 437 Cortez, M. I., 369, 370 directive language, 144 counting Dirichlet series, 465 function, 118, 129 Dirichlet’s theorem, 437 Culik II, K., 505, 515 Dirichlet, P. G. L., 436 cycle, 278 discrete hyperplane, 262 cyclic distance, 7 letter, 510 ultrametric, 7 morphism, 510 division algorithm, 38 cylinder, 29, 329, 375 dominant root condition, 67 Downarowicz, T., 347, 349, 366 D0L Drmota, M., 493, 496, 500, 503 ω-equivalence problem, 506 Duchêne, E., 161

General index 603

Dumont–Thomas numeration, episturmian, 407 254, 319, 502, 513 equicontinuous, 347 Durand, F., 149, 335, 337, 338, equivalent norms, 187 344, 345, 348, 369, 370 equivalent orbit, 354 dynamical system erasing morphism, 10 Bratteli–Vershik, 329 ergodic, 30, 377 Cantor, 325, 329 Birkhoﬀ theorem, 30, 377, 486 conjugacy, 30 individual ergodic theorem, 30, induced, 325 377, 486 measure-theoretic, 30, 376 theorem, 30, 377, 486 stability, 537 Euler–Lagrange’s theorem, 430 symbolic, 28, 53, 374 evaluation map, 26, 37 topological, 377 eventually periodic word, 8, 164 equicontinuous, 347 Evertse, J.-H., 449 topological isomorphism, 30 everywhere-growing endomorphism, 199 edge, 325 exceptional preﬁx, 172 maximal, 329 exduction, 397 minimal, 329 expansion Ehrenfeucht, A., 163, 201, 209, p-, 26, 39 509, 511 base-b, 412 Ehrenfeucht, Lee and Rozen- p q -, 86 berg’s theorem, 209 expansive dynamical system, 345, Ei, H., 527 399 eigenfunction expansive morphism, 199 2 L (µ), 364 exponential language, 136 continuous, 364 exponentially diverging endomor- eigenvalue phism, 200 2 L (µ), 364 exponentially growing word, 194 continuous, 364 extension, 171 Perron–Frobenius, 23 extension type, 172 Eisiedler, M., 445 extraction, 381 elementary morphism, 509 extremal norm theorem, 560 endomorphism, 10, see substitu- extremal number, 439 tion everywhere-growing, 199 factor, 6, 163, 325 exponentially diverging, 200 centric, 197 polynomially diverging, 200 complexity, 163 quasi-uniform, 199 map, 325 entropy, 358 special, 171 topological, 60, 169 factor graph, 176

604 General index

factorial language, 15 q-additive, 454 Fekete’s lemma, 169, 532 q-multiplicative, 454 Fekete, M., 532 complexity, 118 Ferenczi, S., 342, 365, 366, 420 counting, 118 fiber, 148 generating, 96 Fibonacci q-additive, 453 continued fraction, 430 q-automatic, 452 sequence, 417 q-multiplicative, 454 word, 12, 168, 222, 223, 388, q-regular, 453 410 rational, 96, 508 final state, 16 sequential, 102 finite automaton, 17, 414 fundamental lemma, 482 finite difference, 173 Furstenberg, H., 396 finiteness property for capacity, 562 Gambaudo, J.-M., 370 for joint spectral radius, 550 gap, 7 geometric, 275 Gauss Lemma, 38 property (F), 60, 321 Gelfand, I. M., 187 weak (W), 321 Gelfond, A. O., 499 first entrance time map, 325 genealogical order, 9 first finite difference, 173 generalised spectral radius, 532 Fischler, S., 443 fixed point, 10 generalised spectral subradius, 534 folded β-expansion, 77 Fomin, S. V., 349 generating function, 96 Forni, G., 366 generation of bispecial factors, Forrest, A. H., 336 236 fractional part, 49 generic, 380 Fraenkel, A. S., 27, 159, 161 geometric strong coincidence, 293 frequency, 9 GIFS, 255 Fretlöh,D., 319 substitution, 265 Frid, A., 404 Giordano, T., 351, 354 Frougny, Ch., 27, 64, 71, 73, 75, Gjerde, R., 349, 350 77, 111, 116, 128, 153, 160 Glasner, E., 354 full shift, 28, 395 Golden Ratio, 12, 436 function Goldwurm, M., 159 additive, 162 Grabner, P. J., 162, 463, 474 block-additive, 472 graph block-multiplicative, 472 overlaps, 300 co-sequential, 102 prefix-suffix, 256 completely two-piece ancestor, 282

General index 605

graph-directed iterated function immortal letter, 246 system, 255 incidence matrix, 22, 191, 200 greedy of a Bratteli diagram, 326 β-expansion, 56 independence condition, 396 algorithm, 38, 49 indicator function, 6, 356 Grillenberger, C., 169, 391 individual ergodic theorem, 30, group ordered, 351 377, 486 growing word, 198 induced dynamical system, 325 Haar measure, 486 map, 325, 398 Hahn, F., 391 infinite word half-coincidence overlap, 298 automatic, 214 Hamming weight, 47 q-automatic, 214 Handelman, D., 357, 358, 360 Champernowne, 165 Hansel, G., 27, 116, 148, 149, 153, D0L, 506 160 Fibonacci, 168, 222, 223 Harju, T., 161, 505, 515, 529 HD0L, 508 Hausdorff dimension, 445 left-infinite, 180 HD0L lexicographically shift maxi- language, 13 mal, 54 sequence, 507 lsm-word, 54 system, 149, 507 morphic, 181 prolongable, 507 paperfolding, 182 Hedlund, G. A., 163, 348 period-doubling, 244 height, 253 periodic, 164 Heinis, A., 220 purely morphic, 181 Herman, R. H., 324, 330, 333, 335 recurrent, 177, 186 Hollander, M., 68, 69, 111, 272, sparse, 179 321 Sturmian, 167, 177, 223 Holton, C., 335, 366 Thue–Morse, 224 homomorphism of monoids, 4 inflation factor, 23 Honkala, J., 160, 505, 508, 526, injective morphism, 183 529 integer Horner scheme, 39 multiplicatively Host, B., 337, 338, 344, 345, 364, dependent, 27 365, 369 independent, 27 hyperplane representation, 86 discrete, 262 integral part, 49 stepped, 262 internal alphabet, 214 interval exchange i.d.o.c. property, 392 Cantor version, 350

606 General index

map, 391 Kakutani–Rokhlin partition, 401 transformation, 349 Kärki, T., 150 invariance Kátai, I., 488 under similarity, 535 Katok, A., 392, 445 invariant Katznelson, Y., 391 measure, 30 Keane, M., 349, 393 subset, 377 kernel irreducible k-kernel, 147 matrix, 23 q-kernel, 452 morphism, 23 S-kernel, 148 permutation, 392 Khintchine, A. Ya., 430 set of matrices, 536 Kleene star, 14 substitution, 23 Kleene, S., 18 isomorphism, 325 Kozyakin, V. S., 539, 560 measure-theoretic, 31 Krieger, D., 161 topological, 30, 325 Krieger, W., 381 iterated function system Kronecker’s theorem, 264 graph-directed, 255 Kubilius model, 481 Ito, S., 319, 527 Kubilius, J., 438 Iwanik, A., 366 Kwapisz, J., 311, 319 Kwiatkowski, J., 351 Jacobs, K., 349 K˚urka, P., 345, 348 Jewett, R., 381 Johansen, O., 349, 350 Lacroix, L., 366 joint spectral radius, 533 Lagarias, J. C., 532, 551 capacity, 548 Lang, S., 361 computation, 552 language, 13, 341, 374 finiteness property, 550 adherence, 152 introduction, 531 context-free, 89 partition function, 541 D0L, 13 repetition-free words, 543 directive, 144 undecidability, 540 exponential, 136 joint spectral subradius extendable, 374 definition, 533, 534 factorial, 15, 374 Jordan normal form, 187 finite, 14 Jungers, R., 538 HD0L, 13 infinite, 14 k-kernel, see q-kernel of an infinite word, 164 Kaboré,I., 403 polynomial, 136 Kakutani equivalent, 335 prefix-closed, 15 Kakutani, S., 330 ray, 96

General index 607

slender, 15, 134, 160, 525 logarithmically syndetic set, 227 sparse, 136 looping morphism, 517 suffix-closed, 15 Loraud, N., 67, 111 with bounded growth, 15, 96 Lothaire, M., 3, 349, 509 Larcher, G., 464 Luca, F., 423 Lecomte, P., 137, 152 Lee, J.-Y., 319 Maass, A., 347, 348, 369, 370 Lee, K. P., 163, 201, 209 Maes, A., 137, 148, 151 left Mahler, K., 423, 449 extension, 171 Manhattan norm, 2, 191 special factor, 171 map valence, 171 abelianisation, 6 left-infinite word, 180 factor, 325 length, 4 first entrance time, 325 Lenz, D., 343 induced, 325 Leroux, J., 161 lexicographic, 329 Lesigne, E., 389 range, 326 letter, 3 source, 326 cyclic, 510 Vershik, 329 letter-to-letter Markov compacta, 324 morphism, 10, 182 Masur, H., 393 transducer, 20 matrix level, 330 adjacency, 22 Lévy metric, 488 incidence, 22 lexicographic irreducible, 23 map, 329 primitive, 23 order, 9, 48, 186, 328 Mauduit, Ch., 148, 161, 365, 420 Liardet, P., 162 measure invariant, 30 Lindenmayer systems, 149 measure-theoretic Lindenmayer, A., 505 dynamical system, 30 Lindenstauss, E., 445 factor, 31 linearly recurrent isomorphism, 31 Cantor dynamical system, 347 Medynets, K., 350, 351 Linna, M., 161, 509, 529 Mellin–Perron summation for- Liouville’s inequality, 415 mula, 466 Liouville, J., 415 Mendès France, M., 502 Littlewood’s conjecture, 444 Michaux, C., 160 Livshits, A. N., 309, 340 Miller, A., 161 local automaton, 162 Miller, G. A., 96 locally finite, 259 minimal, 375 logarithmic density, 471 dynamical system, 29

608 General index

word, 7 mortal letter, 246 mirror, 4, 446 Mossé, B., 365 formula, 429 multi-graph, 110 Modified Division algorithm, 85 multi-scale quasiperiodic, 408 monoid, 4 multiple tiling, 259 free, 4 multiplicative function, see q- morphism, 4 multiplicative of matrices, 531 multiplicatively Monteil, T., 383, 384 dependent integers, 27 Moody, R.V., 319 independent integers, 27 morphic word, 11, 181 multiplicity morphism, 10, see also endomor- of a bispecial factor, 172 phism Méla, X., 405 cyclic, 510 elementary, 509 natural coding, 392 erasing, 10 neutral bispecial factor, 173 expansive, 199 Nicolay, S., 161 fixed point, 10 Nivat, M., 152 growing, 517 Nogueira, A., 365, 366 injective, 183 non-algebraicity, 539 invertible, 527 non-defective, 560 irreducible, 23 non-erasing morphism, 10 letter-to-letter, 10, 182 non-periodic word, 8 loop-free, 517 non-transient letter, 246 looping, 517 non-trivial morphism, 511 non-erasing, 10 norm, 186 non-trivial, 511 submultiplicative, 25, 531 of ordered groups with order normal unit, 351 matrices, 554 Pisot, 24 number, 413 primitive, 23 normalisation, 26, 71, 112, 160 prolongable, 10, 11 normaliser, 43 proper, 338 Novikov, P. S., 33 read on a Bratteli diagram, 328 Nowakowski, R. J., 161 simplifiable, 509 number uniform, 10 Parry, 58 unit, 25 Perron, 104 Morse and Hedlund theorem, 166, Pisot, 24, 35, 115 175, 199 Salem, 104 Morse substitution, 340 triangular, 117 Morse, M., 163, 224, 340, 348 numeration

General index 609

scale, 65 half-coincidence, 298 numeration system Oxtoby, J., 381 p-ary, 26 abstract, 114 palindrome, 4, 434 adic, 115 palindromic density, 443 canonical (CNS), 78 Pansiot’s theorem, 185 Dumont–Thomas, 319, 502 Pansiot, J.-J., 161, 185, 508, 509, linear, 67 529 positional, 110 paperfolding word, 182 scale, 65, 110 Parikh mapping, 6 numerical value, 26 Parikh vector, 191 S-numerical value, 114 Parry number, 58 Parry, W., 58 occurrence, 341 partial quotients, 428 odometer, 162, 336, 401, 486, 501 partition base, 336 clopen, 330 ω-equivalent Kakutani–Rohlin, 330 D0L systems, 506 KR, 330 HDOL systems, 149 patch, 260 one-sided shift, 28 ancestor, 281 orbit, 374 equivalent, 260 equivalent, 354 path, 16 strongly, 354 Bratteli diagram, 329 of a word, 28 coﬁnal, 329 order label, 16 genealogical, 9 space, 329 lexicographic, 9, 328 successful, 16 radix, 9 P˘aun, G., 134 unit, 351, 352 Pell equation, 137 ordered, 327 period, 8 Bratteli diagram, 327 period cycle, 164 group, 351 period-doubling word, 244 properly, 329 periodic word, 8, 164 ordering Perron number, 104 consecutive, 358 Perron–Frobenius ordinary bispecial factor, 173, 234 eigenvalue, 23 Oseledec, V. I., 391 normalised eigenvector, 24 overlap, 297, 419, 543 theorem, 23, 250 graph, 300 Petersen, K., 405 coincidence, 298 Peth¨o, A., 83, 84 equivalent, 298 p-expansion, 26, 39

610 General index p q -expansion, 86 property (F), see finiteness prop- Pisot erty conjecture, 272 pumping lemma, 18, 89 morphism, 24 purely number, 24, 35, 115 morphic word, 11, 181 unit, 24 substitutive word, 11, 505 substitution, 24 Putnam, I. F., 324, 330, 333, 335, Vijayaraghavan number, 24 351, 354 polynomial language, 136 Pytheas Fogg, N., 509 polynomially bounded word, 194 polynomially diverging endomor- q-additive function, 453 phism, 200 completely, 454 positional numeration system, q-automatic function, 452 110 q-automatic sequence, 452 positive cone, 351 q-kernel, 452 positive uniform frequencies, 390 q-multiplicative function, 454 power of a set, 535 completely, 454 powers of two, 221 q-regular function/sequence, 453 prefix, 6 quasi-greedy expansion, 58 -closed language, 15 quasi-recurrent word, 506 exceptional, 172 quasi-uniform endomorphism, proper, 6 199 unioccurrent, 171 Queffélec, M., 309, 434 prefix-suffix graph, 256 preperiod, 8, 164 Rabin, M. O., 17 primitive radix matrix, 23 order, 9, 39 morphism, 23 point, 48 substitution, 23 Rampersad, N., 161 word, 164, 178 range map, 326 problem rank D0L ω-equivalence, 506 topological, 347 D0L prefix, 509 infinite, 347 HD0L ω-equivalence, 149 ranking, 159 prolongable Rao, H., 319, 527 D0L system, 506 rational function, 96, 508 morphism, 10 Ratner, M., 406 proper Rauzy substitution, 338 fractal, 253 properly graph, 176, 384 ordered, 329 Rauzy, G., 176, 248, 338, 384

General index 611

Ravikumar, B., 161 Rudin–Shapiro word, 388 recurrence Rudolph, D. J., 366, 406 linear, 408 Ruohonen, K., 508 uniform, 375 recurrent word, 7, 171, 177, 186 s-adic construction, 226 reducible S-recognisable set, 117 set of matrices, 536 Séébold, P., 8 redundancy transducer, 44 Sadun, L., 321 regular function/sequence, see q- Sakarovitch, J., 77, 128 regular Salem number, 104 regular language, 88 Salomaa and Soittola’s theorem, relatively dense, 260 191 repetition, 419, 543 Salomaa, A., 134, 191, 509 repetitive, 260 Salon, O., 150 representation sandwich set, 180 U-, 65, 109 sandwich set theorem, 188 S-, 114, 117 Sataev, E., 397 p-ary, 26, see expansion scale, 65, 110 BV, 331 scaling property, 535 greedy, 26 Schützenberger, M.-P., 33 integer, 86 Schlickewei, H. P., 450 return word, 178, 341 Schmidt’s subspace theorem, 422 reversal, 4 Schmidt, K., 59, 158 Ridout’s Theorem, 420 Schmidt, W. M., 434, 438 Ridout, D., 420 Scott, D., 17 right second finite difference, 173 extension, 171 seed patch, 276, 282 special factor, 171 self-replicating multiple tiling, valence, 171 271 right transducer, 21 semigroup, 4 right context, 88 of matrices, 531 Rigo, M., 137, 148, 150, 158, 317, sequence 526 2-regular, 545 Rohlin, V. A., 330 q-automatic, 19, 138, 452 Rosenthal, A., 381 q-regular, 453 Rota, G. C., 531, 536 S-automatic, 138 Roth’s Theorem, 416 sequential, 102 Roth, K. F., 416 transducer, 20 Roy, D., 438 set Rozenberg, G., 163, 201, 209, 509, Cantor, 7 511 eventually periodic, 8

612 General index

recognisable, 27, 109 SRS, see shift radix system S-recognisable, 117 stammering, 419 U-recognisable, 73, 110 state, 362 syndetic, 7 accessible, 16 Shallit, J., 20, 67, 111, 161, 508, co-accessible, 16 512 final, 16 shape of a Rauzy graph, 178 terminal, 16 shift, 28, 139, 374 stationary, 337 circular, 170 Bratteli diagram, 337 full, 28 Steiner, W., 137, 158 one-sided, 28 stepped hyperplane, 262 two-sided, 28 strand shift radix system, 83 basic formal, 291 shuffle, 129 basic geometric, 253 Siegel, A., 249 formal, 291 simple geometric, 291 Bratteli diagram, 327 Strang, G., 531, 536 hat, 327 strictly ergodic, 390 simplifiable morphism, 509 strong bispecial factor, 172 Sinai, Y. V., 349 strong mixing, 408 Sing, B., 319 strongly orbit equivalent, 354 sink, 18 Sturmian Sirvent, V., 296 expansion, 421 Skau, C. F., 324, 330, 333, 335, subshift, 348 337, 338, 344, 345, 351, 354 word, 8, 167, 177, 223, 388 S-kernel, 148 subadditive function, 168 skew-product, 498 subadditivity, 532 slender language, 15, 134, 160, submultiplicative norm, 25, 531 525 subshift, 28, 53, 374 SOE, see strongly orbit equiva- aperiodic, 28, 337, 385 lent coded, 54 Soittola, M., 191 conjugate, 30 Solomyak, B., 71, 77, 153, 272, entropy, 60 296, 319, 370 finite type, 28, 53, 395 source map, 326 periodic, 28, 337 sparse language, 136 sofic, 28, 54, 395 sparse word, 179 substitution, 343 special factor, 171 Toeplitz, 349 bound on their number, 219 substitution, 10, see endomor- spectral radius, 25, 187, 512 phism, see morphism square, 4 Chacon, 342

General index 613

GIFS, 265 terminal state, 16 invertible, 527 theorem irreducible, 23 Cobham, 181 Morse, 340 Ehrenfeucht, Lee and Rozen- Pisot, 24 berg, 209 irreducible, 25 Grillenberger, 169 reducible, 25 Kronecker, 264 primitive, 23 Morse and Hedlund, 166, 175, proper, 338 199 read on a stationary Bratteli di- Pansiot, 185 agram, 337 Perron–Frobenius, 23, 250 subshift, 343 Salomaa and Soittola, 191 unit, 25 sandwich set, 188 substitutive Theys, J., 534, 539 word, 11 Thomas, W., 129, 148 word sequence, 507 Thue, A., 33, 224 subword Thue–Morse complexity, 163 Bratteli diagram, 340 scattered, 6 continued fraction, 434 successor, 128 word, 12, 224, 388, 410, 560 suﬃx, 6 Thurston, W., 248 -closed language, 15 Thuswaldner, J., 83, 84 proper, 6 Tichy, R. F., 162 Sugisaki, F., 360 tiles, 259 sum-of-digits, 162, 452, 457, 467, tiling, 162, 259 499, 502 synchronisation, 298 switched linear system, 538 lattice multiple, 320 symbol, 3 multiple, 259 symbolic dynamical system, 28 patch, 260 synchronisation lemma, 221, 229 property, 272 syndetic, 227 self-replicating multiple, 271 set, 7 self-similar, 261 system tip, 263 D0L, 13, 149, 505 Toeplitz nearly primitive, 507 subshift, 349 prolongable, 506 word, 170, 349 HD0L, 149, 507 topological conjugacy, 30 tag sequence, 138 dynamical system, 29 telescoping, 326, 328 entropy, 60, 169, 390 Tenenbaum, G., 457 isomorphism, 30

614 General index

total irrationality, 392 substitution, 25 tower, 330 universal counter-example, 179 level, 330 Transcendence criterion, 423 valence, 171 transducer, 20 Veech, W., 393, 396 co-sequential, 102 Vershik map, 329 digit-conversion, 41 Vershik, A. M., 324, 329, 330, letter-to-letter, 20, 101 340, 399 right, 21 vertex, 325 sequential, 20, 102 Villemaire, R., 160 transformation β-, 56, 162 Walters, P., 357, 358 adic, 324 Wang, Y., 538, 551 transient part, 164 weak (W), see finiteness property transition weak bispecial factor, 172 function, 17 weight, 47, 475 relation, 16 weight function, 379 structure, 141 on a graph, 393 triangular Weiss, B., 354 matrices, 536 Wen, Z., 527 joint spectral radius of, 554 width of a morphism, 11 number, 117 Wirsing, E., 451 trie, 116 word, 3, see sequence trim, 16 automatic, 19 S Tsitsiklis, J. N., 539, 540 -automatic, 138 Turán–Kubilius inequality, 484 β-admissible, 57 two-piece ancestor graph, 282 bi-infinite, 5 bounded, 198 U-recognisable, 73, 110 Chacon, 24, 342 U-representation, 65 Champernowne, 148, 395, 413 underlying input automaton, 20 characteristic, 5 uniform Diophantine approxima- comparable, 522 tion, 437 concatenation, 4 uniform frequencies, 376 D0L infinite word, 506 uniformly recurrent word, 7 distance, 7 uniformly discrete, 260 empty, 4 unioccurrent prefix, 171 eventually periodic, 8 unique ergodicity, 30, 32, 380 exponentially growing, 194 unit, 24 factor, 6 morphism, 25 Fibonacci, 12, 410 order, 351, 352 growing, 198

General index 615

infinite lexicographically shift maximal, 54 lsm-word, 54 one-sided, 5 two-sided, 5 length, 4 minimal, 7 mirror, 4 morphic, 11 purely, 11 nearly periodic, 515 non-periodic, 8 ω-equivalent, 149 period, 8 periodic, 8 polynomially bounded, 194 prefix, 6 preperiod, 8 primitive, 164, 178 purely substitutive, 11, 505 quasi-recurrent, 506 recurrent, 7 reversal, 4 Sturmian, 8 substitutive, 11 purely, 11, 505 subword (scattered), 6 suffix, 6 Thue–Morse, 12, 410, 560 Toeplitz, 349 uniformly recurrent, 7 Wythoff’s game, 161

Yu, S., 13

Zamboni, L., 335, 366 Z-balanced 1-systems, 519 zero automaton, 41, 62, 288 zero spectral radius, 553