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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 K>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<v(lexicographic order), 9 fw (x) (frequency), 376, 380 u ≤p v (u is a prefix of v), 513 Γc (self-replicating translation u v (lexicographic order), 9 set), 263 u ≺ v (radix order), 9 Γe (self-similar translation set), u v (radix order), 9 261 L≤n (concatenation of at most n [γ,i]∗ (tip), 263 words in L), 14 ∗ → k U [γ,i]g (projected face), 263 L[n n ] (language s.t. (n)= k Gµ,f , 379 n ), 129 Gn (Rauzy graph), 176 L(A) (language recognised by A), Gn (X) (Rauzy graph), 384 16 Gn (x) (Rauzy graph), 384 L(X) (language), 374 GO (graph of overlaps), 300 L(x) (language), 7 © 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 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 © 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 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 alpha- bet 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 © 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 598 Notation index ζ(s) (Riemann zeta function), 459 ζf (Fibonacci continued frac- tion), 411 ζa (real number whose continued fraction expansion is given by the word a), 430 ζt (Thue–Morse continued frac- tion), 411 ζ(s, α) (Hurwitz zeta function), 459 © 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 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 B¨uchi, 54 transformation, 324 complete, 17 adjacency matrix, 22 deterministic, 17 Adjan, S.
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