OOn a Family of Random Noble Means Substitutions
Markus Moll
Fakult¨at f¨ur Mathematik, Universit¨at BielefeldUniversit¨atsstraße 25, D–33615 Bielefeld, Germany
Abstract
In 1989, Godr`eche and Luck [1] introduced the concept of local mixtures of primitive substitution rules along theexample of the well-known Fibonacci substitution and foreshadowed heuristic results on the topological entropy andthe spectral type of the diffraction measure of associated point sets. In this contribution, we present a generalisation ofthis concept by regarding the so-called ‘noble means families’, each consisting of finitely many primitive substitutionrules that individually all define the same two-sided discrete dynamical hull. We report about results in the randomisedcase on topological entropy, ergodicity of the two-sided discrete hull, and the spectral type of the diffraction measureof related point sets.PACS: 45.30.+s, 61.44.Br
Consider the binary alphabet A = { a , b } . For an arbi-trary but fixed integer m ≥ ≤ i ≤ m , we definethe noble means substitution (NMS) rule ζ m,i : A → A ∗ by ζ m,i : (cid:26) a (cid:55)→ a i ba m − i , b (cid:55)→ a , where M m := (cid:18) m
11 0 (cid:19) is its (unimodular) substitution matrix. The family N m := (cid:8) ζ m,i | m ∈ N , ≤ i ≤ m (cid:9) is called a noble means family and each of its membersis a primitive Pisot substitution with inflation multiplier λ m := ( m + √ m + 4) / λ (cid:48) m :=( m − √ m + 4) /
2. The two-sided discrete (symbolic) hull X m,i of ζ m,i is defined as the orbit closure of a fixed pointin the local topology. Each X m,i is reflection symmetricand aperiodic in the sense that it does not contain anyperiodic element. For fixed m ∈ N , one observes that the ζ m,i are pairwise conjugate and therefore all individual X m,i coincide; see [2, Ch. 4] for background.Now, we fix m ∈ N and a (strictly positive) probabilityvector p m = ( p , . . . , p m ) and define a random substitu-tion ζ m on A by ζ m : a (cid:55)→ ζ m, ( a ) with probability p , ... ... ζ m,m ( a ) with probability p m , b (cid:55)→ a . We refer to ζ m as a random noble means substitution(RNMS) . Both the substitution matrix and the inflation ∗ email: [email protected] multiplier are the same as in the NMS case. We aim atthe local mixture of all members of N m , which means thatwe independently apply ζ m to each letter of some word w ∈ A Z . In this case, the two-sided discrete stochas-tic hull X m is defined as the smallest closed and shift-invariant subset of A Z with the property that X m ⊂ X m ,where X m := (cid:110) w ∈ A Z | w is an accumulationpoint of (cid:0) ζ km ( a | a ) (cid:1) k ∈ N (cid:111) . Both X m,i and X m are completely characterised by thelegal subwords. Here, a word w ∈ A ∗ is ζ m -legal if thereis a k ∈ N such that w is a subword of at least one reali-sation of the random variable ζ km ( b ). The set of ζ m -legalwords of length (cid:96) is henceforth denoted by D m,(cid:96) . Onecan show that X m,i (cid:40) X m by considering the subword bb and that the system ( X m , S ), where S denotes the shift, istopologically transitive but not minimal. Note that X m isinvariant under alterations of p m as long as p m is strictlypositive. For m ∈ N and n ≥
3, the set of exact RNMS words isgiven by G m,n := m (cid:91) i =0 m (cid:89) j =0 G m,n − − δ ij , (1)where G m, := { b } , G m, := { a } and δ ij denotes the Kro-necker function. The product in Eq. (1) is understood viaconcatenation of words. Now, assume that p m is strictlypositive. The complexity function C m : N → N , (cid:96) (cid:55)→ |D m,(cid:96) | a r X i v : . [ m a t h . D S ] D ec R × R RZ [ λ m ] L m Z [ λ m ] L L (cid:63) π π − − (cid:63) ⊂ ⊂ ⊂ densedense Figure 1:
Cut and project scheme for the noble meanssets Λ m,i . of ζ m is unknown, but the knowledge of the exact RNMSwords is enough [3] to compute the topological entropy H m of ζ m for any m ∈ N to be H m = lim n →∞ log (cid:0) C m ( (cid:96) m,n ) (cid:1) (cid:96) m,n = lim n →∞ log (cid:0) |G m,n | (cid:1) (cid:96) m,n = λ m − − λ (cid:48) m ∞ (cid:88) i =2 log (cid:0) m ( i −
1) + 1 (cid:1) λ im > , where (cid:96) m,n is the length of any word w ∈ G m,n . Thenumerical values of H m for 1 ≤ m ≤ Table 1:
Numerical values of H m for 1 ≤ m ≤ m H m . . . . H m > H m +1 for all m ∈ N and H m m →∞ −−−−→ ( X m , S ) The known concept of the induced substitution [4, Ch.5] that acts on the alphabet of legal subwords of a fixedlength can be generalised to the stochastic setting of theRNMS case. One obtains a random substitution rule( ζ m ) (cid:96) : D m,(cid:96) → D ∗ m,(cid:96) and one can prove that the inducedsubstitution matrix M m,(cid:96) is a primitive matrix which en-ables the application of Perron–Frobenius (PF) theory.For fixed m , (cid:96) ∈ N , let w ∈ D m,(cid:96) be a ζ m -legal word.We define a shift-invariant probability measure µ m on thecylinder sets Z k ( w ) = { v ∈ X m | v [ k,k + (cid:96) − = w } for any k ∈ Z by µ m (cid:0) Z k ( w ) (cid:1) := R m,(cid:96) ( w ) , (2)where R m,(cid:96) ( w ) is the entry of the (statistically nor-malised) right PF eigenvector of M m,(cid:96) according to theword w . Theorem 1 ([5, 6]) . Let X m ⊂ A Z be the hull of therandom noble means substitution for m ∈ N and µ m theshift-invariant probability measure of Eq. (2) on X m . Forany f ∈ L ( X m , µ m ) and for an arbitrary but fixed s ∈ Z , lim N →∞ N N + s − (cid:88) i = s f ( S i x ) = (cid:90) X m f d µ m Figure 2:
The strip R × (cid:83) mi =0 W m,i (light) is strictlyincluded in R × W m (dark). Here, this is illustrated for m = 2. holds for µ m -almost every x ∈ X m . Theorem 1 implies that µ m is ergodic. The proof canbe accomplished via an application of Etemadi’s formula-tion of the strong law of large numbers [7] and a suitablereorganisation of the summation over the characteristicfunction of some cylinder set. The geometric realisations Λ m,i of fixed points of eachnoble means substitution can be derived as regular modelsets within the cut and project scheme ( R , R , L m ), where L m := { ( x, x (cid:48) ) | x ∈ Z [ λ m ] } ; see Figure 1. Here, theletters a and b are identified with closed intervals of length λ m and 1, respectively, and the left endpoints are chosenas control points. The windows W m,i for Λ m,i , in thegeneric cases 0 < i < m , are W m,i := iτ m + [ λ (cid:48) m ,
1] with τ m := − m ( λ (cid:48) m + 1) , while in the singular cases i = 0 and i = m , one finds W ( a | a ) m, := [ λ (cid:48) m , , W ( a | b ) m, := ( λ (cid:48) m , ,W ( a | a ) m,m := ( − , − λ (cid:48) m ] , W ( b | a ) m,m := [ − , − λ (cid:48) m ) , distinguished according to the legal two-letter seeds.Now, one can prove that each member of the continu-ous RNMS hull Y m is a relatively dense subset of an el-ement of the LI class of the model set Θ( W m ), withinthe cut and project scheme of Figure 1, with window W m = [ λ (cid:48) m − , − λ (cid:48) m ] and therefore a Meyer set by [8,Thm. 9.1]. The volume of the interval W m is minimalwith this property, and it strictly contains (cid:83) mi =0 W m,i ; seeFigure 2 for an illustration in the case of m = 2.Consequently, each geometric realisation of a randomnoble means word is a naturally arising instance of aMeyer set with entropy. The diffraction of the NMS cases is well understood dueto their characterisation as regular model sets [2, Ch. 9]2 .0 4.0 6.0 8.0 10.00.10.20.30.40.5
Figure 3:
The pure point (light) and absolutely contin-uous (dark) part of (cid:98) γ with p = (1 / , /
2) is illustrated. whereas the results presented in [9, 10], suggest the pres-ence of a continuous part in the diffraction spectrum inthe RNMS case.Because of Theorem 1, the suspension [11, 12] ν m of µ m on Y m leads to a continuous and ergodic dynamicalsystem ( Y m , R , ν m ). Now, let δ Λ := (cid:80) x ∈ Λ δ x be the Diraccomb for a random noble means set Λ ∈ Y m . One cancompute the autocorrelation of δ Λ to be ν m -almost surelygiven by γ := lim R →∞ δ Λ R ∗ (cid:103) δ Λ R vol( B R ) = E ( δ Λ (cid:126) (cid:102) δ Λ ) , where Λ R := Λ ∩ B R (0) and (cid:126) denotes the volume-averaged convolution by balls. The diffraction measureis given by the Fourier transform of γ and reads (cid:98) γ = lim R →∞ B R ) | E ( (cid:100) δ Λ R ) | + lim R →∞ B R ) V ( (cid:100) δ Λ R ) , where E and V refer to mean and variance with respect tothe measure ν m . Now, the following two key propertiesfinally lead to an explicit expression for (cid:98) γ . • It is enough to study (cid:98) γ on the basis of exact RNMSwords, as defined in Eq. (1), because ζ m -legality ofa word w ∈ A ∗ means that w is a subword of a wordin G m,n for a suitably chosen n ∈ N . • It is not difficult to prove that G m,n = (cid:8) w ∈ A ∗ | w = ζ n − m ( b ) (cid:9) (3)and even more that the two stochastic processes,based on the substitution rule and the concatena-tion rule, are equal. Note that the equality in Eq.(3) means that there is at least one realisation of therandom variable ζ n − m ( b ) that equals w .For convenience, we restrict to m = 1 in the followingand define for n ≥ X n ( k ) by X n ( k ) := (cid:40) X n − ( k ) + e − π i kλ n − X n − ( k ) , (cid:104) p (cid:105) ,X n − ( k ) + e − π i kλ n − X n − ( k ) , (cid:104) p (cid:105) , Figure 4:
Approximation of the diffraction measure for m = 1 and with p = (1 / , /
2) is illustrated. with X ( k ) = e − π i k and X ( k ) = e − π i kλ . Here, X n ( k )corresponds to exact RNMS words in G ,n +1 . Therefore,we consider averaging over the sequence L n = λ n andfind the following result. Proposition 2 ([5]) . For any n ∈ N , consider the func-tion φ n : R → R + , defined by φ n ( k ) := 1 L n V (cid:0) X n ( k ) (cid:1) . The sequence ( φ n ) n ∈ N converges uniformly to the contin-uous function φ : R → R + , with φ ( k ) := 2 p p λ √ ∞ (cid:88) i =2 λ − i Ψ i ( k ) . (4)Here, Ψ n : R → R + is a bounded and smooth functionthat monotonically decreases in n , defined byΨ n ( k ) := 12 (cid:12)(cid:12) (1 − e n − ) E n − − (1 − e n − ) E n − (cid:12)(cid:12) , where e n := e − π i kλ n and E n := E ( X n ( k )). This fixes theabsolutely continuous part of (cid:98) γ . The pure point part canbe computed via the recursion relation E n = ( p + p e n − ) E n − + ( p + p e n − ) E n − , (5)where E := e − π i k and E := e − π i kλ . This yields (cid:98) γ ( { k } ) = lim n →∞ L n | E ( X n ( k )) | and an approximation of ( (cid:98) γ ) pp and ( (cid:98) γ ) ac is illustratedin Figure 3 together with a sketch of the full diffraction,based on the recursion of Eq. (5) with n = 6, in Figure4. Considering the Lebesgue decomposition (cid:98) γ = ( (cid:98) γ ) pp +( (cid:98) γ ) ac + ( (cid:98) γ ) sc , we find that (cid:98) γ = ( (cid:98) γ ) pp + φ ( k ) λ, where λ denotes the Lebesgue measure and φ the densityfunction of Eq. (4).3 . − . − . . . . . . . . . Figure 5:
Distribution of control points generated by a (dark) and b (light) in the internal space in the case of p = (1 / , / ζ ( b )(i.e. 2178309 points) to the internal space. It is possible to compute the pure point part from therecursion relation in Eq. (5). Another interesting ap-proach comes from the theory of iterated function sys-tems and inflation-invariant measures. Here, one findsthat ( (cid:98) γ ) pp = (cid:88) k ∈L (cid:126) | (cid:98) η a ( − k (cid:48) ) + (cid:98) η b ( − k (cid:48) ) | δ k , where L (cid:126) = π ( L ∗ ) = Z [ λ ] / √
5, with L ∗ the dual latticeof L , is the Fourier module. In the following, we write ξ := λ (cid:48) . The invariant measures (cid:98) η a , (cid:98) η b can be approxi-mated via the recursion relation (cid:18) (cid:98) η a ( k ) (cid:98) η b ( k ) (cid:19) = | ξ | n (cid:16) n (cid:89) (cid:96) =1 p A (cid:96) ( k ) + p B (cid:96) ( k ) (cid:17) (cid:18) (cid:98) η a ( kξ n ) (cid:98) η b ( kξ n ) (cid:19) , where the matrices A (cid:96) ( k ) and B (cid:96) ( k ) are given by (cid:18) e − π i kξ (cid:96) −
11 0 (cid:19) and (cid:18) − π i kξ (cid:96) (cid:19) . As ξ n → n → ∞ , an appropriate choice of theeigenvector (cid:0) (cid:98) η a (0) , (cid:98) η b (0) (cid:1) T for the equation (cid:18) (cid:19) (cid:18) (cid:98) η a (0) (cid:98) η b (0) (cid:19) = λ (cid:18) (cid:98) η a (0) (cid:98) η b (0) (cid:19) fixes the recursion. Since (cid:98) η a (0) + (cid:98) η b (0) must be the pointdensity of some random golden means set, which always is λ / √
5, one finds (cid:98) η a (0) = 1 / √ (cid:98) η b (0) = ( λ − / √ a and b , respectively, is illustrated inFigure 5. Acknowledgement
The author wishes to thank Michael Baake, TobiasJakobi and Johan Nilsson for helpful discussions. Thiswork is supported by the German Research Foundation (DFG) via the Collaborative Research Centre (CRC 701)through the faculty of Mathematics, University of Biele-feld.
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