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a one - sided infinite sequence @xmath1 over the alphabet @xmath2 is called a ( classical ) _ kolakoski sequence _ ( named after w.  kolakoski who introduced it in 1965 , see  @xcite ) , if it equals the sequence defined by its run lengths , e.g. : @xmath3 here , a _ run _ is a maximal subword consisting of identical letters . the sequence @xmath4 is the only other sequence which has this property . one way to obtain @xmath1 of ( [ eq : kol ] ) is by starting with @xmath5 as a seed and iterating the two substitutions @xmath6 alternatingly , i.e. , @xmath7 substitutes letters on even positions and @xmath8 letters on odd positions ( we begin counting at @xmath9 ) : @xmath10 clearly , the iterates converge to the kolakoski sequence @xmath1 ( in the obvious product topology ) , and @xmath1 is the unique ( one - sided ) fixed point of this iteration .    one can generalize this by choosing a different alphabet @xmath11 ( we are only looking at alphabets with @xmath12 ) . such a ( generalized ) kolakoski sequence , which is also equal to the sequence of its run lengths , can be obtained by iterating the two substitutions @xmath13 alternatingly . here , the starting letter of the sequence is @xmath14 . we will call such a sequence a kolakoski-@xmath15 sequence , or kol@xmath15 for short . the classical kolakoski sequence @xmath1 of ( [ eq : kol ] ) is therefore denoted by kol@xmath16 ( and @xmath17 by kol@xmath18 ) . while little is known about the classical kolakoski sequence ( see  @xcite ) , and the same holds for all kol@xmath15 with @xmath14 odd and @xmath19 even or vice versa ( see  @xcite ) , the situation is more favorable if @xmath14 and @xmath19 are either both even or both odd . if both are odd , one can , in some cases , rewrite the substitution as a substitution of pisot type ( see  @xcite ) , which can be described as ( limit ) aperiodic model sets . by this method , kol@xmath20 is studied in  @xcite and shown to be a deformed model set . the case where both symbols are even will be studied below . it is the aim of this article to determine structure and order of the sequences kol@xmath21 . this will require two steps : first we establish an equivalent substitution of constant length for kol@xmath21 and analyze it with methods known from the theory of dynamical systems . then we conclude diffractive properties from this . remark : every kol@xmath15 can uniquely be extended to a bi - infinite ( or two - sided ) sequence . the one - sided sequence ( to the right ) is kol@xmath15 as explained above . the added part to the left is a reversed copy of kol@xmath22 , e.g. , in the case of the classical kolakoski sequence of ( [ eq : kol ] ) , this reads as @xmath23 where `` @xmath24 '' denotes the seamline between the one - sided sequences . note that , if @xmath25 ( or @xmath26 ) , the bi - infinite sequence is mirror symmetric around the first position to the left ( right ) of the seamline . the bi - infinite sequence equals the sequence of its run lengths , if counting is begun at the seamline . alternatively , one can get such a bi - infinite sequence by starting with @xmath27 and applying the two substitutions to get @xmath28 in the first step and so forth . this also implies that kol@xmath15 and kol@xmath22 will have the same spectral properties , and it suffices to study one of them . if both letters are even numbers , i.e. , @xmath29 and @xmath30 ( with @xmath31 , where we can concentrate on @xmath32 by the above discussion ) , one can build blocks of two letters and obtain an ( ordinary ) substitution . setting @xmath33 and @xmath34 , these substitutions and their _ substitution matrix _ @xmath35 ( sometimes called _ incidence matrix _ of the substitution ) are given by @xmath36 where the entry @xmath37 is the number of occurrences of @xmath38 in @xmath39 ( @xmath40 ; sometimes the transposed matrix is used ) . a bi - infinite fixed point can be obtained as follows : @xmath41 this corresponds to the unique bi - infinite kol@xmath21 according to our above convention . a substitution @xmath42 is _ primitive _ if the corresponding substitution matrix @xmath35 is primitive , i.e. , @xmath43 has positive entries only for some @xmath44 . equivalently , @xmath42 is primitive if there exists a positive integer @xmath44 such that every @xmath45 occurs in @xmath46 for all @xmath47 . the vector @xmath48 with components @xmath49 , for @xmath45 , is called the _ length _ of the substitution @xmath42 . if all @xmath50 are equal , @xmath42 is a substitution of _ constant length_. for the substitution @xmath51 of  ( [ eq : subs1 ] ) , we have @xmath52 which is therefore not of constant length ( recall that @xmath53 ) .    we will also need some notions from the theory of _ dynamical systems _ , see  @xcite and ( * ? ? ? * chapters 1 , 5 and 7 ) for details . let @xmath42 be a primitive substitution over @xmath54 and @xmath55 a bi - infinite fixed point of @xmath42 ( i.e. , @xmath56 for some @xmath57 ) . denote by @xmath58 the @xmath59th letter of @xmath60 ( @xmath61 ) and by @xmath62 the ( i.e. , @xmath63 ) . let @xmath54 be equipped with the discrete and @xmath64 with the corresponding product topology . if we set @xmath65 then @xmath66 is a _ dynamical system_. since we require @xmath42 to be primitive , this dynamical system is _ minimal _ ( i.e. , @xmath67 is dense in @xmath68 for all @xmath69 ) , does not depend on the chosen fixed point @xmath60 ( if more than one exists , which is possible in the two - sided situation ) and has a unique probability measure @xmath70 associated with it . in other words , it is _ strictly ergodic_. on the hilbert space @xmath71 , we have the unitary operator @xmath72 if @xmath73 for some @xmath74 , we call @xmath75 an _ eigenvalue _ of @xmath66 and @xmath76 the corresponding _ eigenfunction_. the _ spectrum _ ( of the dynamical system ) is said to be a _ pure point dynamical spectrum _ ( or _ discrete spectrum _ ) , if the eigenfunctions span @xmath71 . if @xmath77 is the only eigenvalue and the only eigenfunctions are the constants , the spectrum is _ continuous_. it is also possible that it has pure point and continuous components . in that case it is called _ partially continuous_. two dynamical systems @xmath78 and @xmath79 are _ isomorphic _ ( or _ measure - theoretically isomorphic _ ) , if there exists an invertible measurable map @xmath80 , almost everywhere defined , such that @xmath81 preserves the measure and the dynamics ( i.e. , @xmath82 ) .    the spectral theory of primitive substitutions of constant length is well understood . by the following criterion , we know that the substitutions @xmath51 of  ( [ eq : subs1 ] ) are related to substitutions of constant length .    [ lem : dek - v1 ] ( * ? ? ? * section v , theorem 1 ) let @xmath42 be a substitution of nonconstant length @xmath48 . if @xmath48 is a right eigenvector of the corresponding substitution matrix @xmath35 , then @xmath83 is isomorphic to a substitution dynamical system generated by a substitution of constant length .    since the substitutions @xmath51 of  ( [ eq : subs1 ] ) fulfillnote that from  ( [ eq : subs0 ] ) , one can also construct a primitive substitution by distinguishing odd and even positions , e.g. , for kol@xmath84 we would get ( we use @xmath85 as mark for even positions ) @xmath86 instead of  ( [ eq : subs1 ] ) , we would get a substitution with substitution matrix @xmath87 which is also primitive ( @xmath88 has positive entries only ) , but does not fulfill the requirements of lemma  [ lem : dek - v1 ] . the eigenvalues of this @xmath35 are @xmath89 . ] the requirements of this lemma , the next task is now to construct the corresponding substitutions of constant length . this is achieved by numbering the @xmath90 s and @xmath91 s in  ( [ eq : subs1 ] ) , i.e. , we make the substitutions @xmath92 , respectively @xmath93 . then , the former substitutions  ( [ eq : subs1 ] ) induce @xmath94 from this we get substitutions of constant length @xmath95 ( the eigenvalue of the substitution matrix @xmath35 in  ( [ eq : subs1 ] ) ) by parting the right sides in blocks of @xmath95 letters . for example , let @xmath96 and @xmath97 . then @xmath98 and one extracts the following substitution of constant length @xmath99 : @xmath100 in the same way , we get substitutions of constant length @xmath95 from  ( [ eq : subs2 ] ) . note that these substitutions are all primitive since ( compare to  ( [ eq : subs2 ] ) ) in every block of @xmath101 successive letters ( note that we use @xmath102 ) every letter of @xmath103 occurs , so if @xmath104 , then @xmath105 has positive entries only ( this holds for @xmath106 ) . note also that we can reduce the alphabet by one letter by identifying @xmath107 ( @xmath108 in the example  ( [ eq : kol42h2 ] ) ) , because both @xmath90 s always yield the same substitution . let us now determine the positions of @xmath109 in the sequence @xmath60 generated by  ( [ eq : subs2 ] ) . they are given by @xmath110 for some @xmath111 ( e.g. , @xmath112 ) . therefore we get @xmath113 . the _ height _ @xmath114 of a primitive substitution @xmath42 of constant length @xmath0 which generates a sequence @xmath60 is defined as @xmath115 then the following lemma holds . [ lem : dek - ii13 ] let @xmath66 be a dynamical system , where @xmath42 is a primitive substitution of constant length @xmath0 and height @xmath114 . then the pure point part of this dynamical system is isomorphic to the dynamical system @xmath116 , where @xmath117 is the addition of @xmath118 on the abelian group @xmath119 , i.e. , the direct product of the @xmath0-adic integers @xmath120 and the cyclic group @xmath121 of order @xmath114 . therefore the pure point dynamical spectrum is given by @xmath122    note that @xmath123 where @xmath124 are the distinct primes dividing @xmath0 , see  ( * ? ? ? * section 3.10 ) . the lemma is just a reformulation of  ( * ? ? ? * theorem ii.13 ) , compare with  ( * ? ? ? * section vi . ) and  ( * ? ? ? * section 7.3 ) . [ prop : spectrum ] suppose @xmath125 has pure point dynamical spectrum , where @xmath51 is the substitution of  ( [ eq : subs1 ] ) . then @xmath126 where @xmath117 is the addition of @xmath118 and @xmath127 the addition of @xmath77 .    for the substitution  ( [ eq : subs2 ] ) of constant length @xmath128 , we have already seen that @xmath129 . therefore , using  ( [ eq : def_height ] ) , the height of this substitution is @xmath5 if @xmath0 is odd and @xmath77 if @xmath0 is even . the dynamical system of the substitution  ( [ eq : subs2 ] ) is isomorphic to @xmath125 by lemma  [ lem : dek - v1 ] , therefore they have the same spectrum . the remaining statement follows from lemma  [ lem : dek - ii13 ] . we want to show that the spectrum of @xmath51 is indeed pure point . for this we use slightly different substitutions of constant length that we deduce from @xmath51 . we substitute @xmath130 and @xmath131 ( so in  ( [ eq : subs2 ] ) we build essentially blocks of two , e.g. , @xmath132 ) . we get @xmath133 which again gives substitutions of constant length @xmath128 . they are all primitive substitutions by the same argument as before ( in every block of @xmath134 successive letters every letter occurs ) . in the case @xmath135 , we can reduce the alphabet by one letter by identifying @xmath136 . so we have two cases , @xmath97 with substitutions @xmath137 and @xmath135 with substitutions @xmath138 ( it is cumbersome to write down such a @xmath138 in general form , but we will investigate its structure in the next section ) . now the height of @xmath138 and @xmath139 is always @xmath77 , because if @xmath135 we get @xmath140 , and if @xmath97 we get @xmath141 . let @xmath42 be a primitive substitution of constant length @xmath0 and height @xmath142 . one says that @xmath42 admits a _ coincidence _ , if there exist a @xmath44 and @xmath143 such that @xmath144 is the same for all @xmath45 ( the @xmath38th letter of each @xmath145 is the same , i.e. , @xmath146 admits a column of identical values ) . [ lem : dekiii7 ] ( * ? ? ? * section iii , theorem 7 ) let @xmath66 be a substitution dynamical system of constant length and height @xmath142 . then @xmath66 has pure point dynamical spectrum if and only if @xmath42 admits a coincidence . if a substitution has height @xmath147 , one gets a substitution of height @xmath77 by combining letters into blocks of @xmath148 letters . if this new substitution has pure point dynamical spectrum , so has the original substitution of height @xmath148 , see  @xcite . obviously , we get the following : if the substitutions @xmath138 and @xmath139 ( which arise from  ( [ eq : subs3 ] ) ) admit coincidences , then the dynamical systems defined by @xmath51 of  ( [ eq : subs1 ] ) have pure point dynamical spectrum . let us first check @xmath139 of  ( [ eq : kol2m2 ] ) for coincidences . for this we begin by exploring the structure : @xmath149 has two @xmath150 s at position @xmath9 and @xmath151 , @xmath152 has an @xmath150 at position @xmath153 , etc . we get an @xmath150 in @xmath154 at position @xmath155 for @xmath156 . similar arguments show that there is an @xmath157 in @xmath154 at position @xmath158 for @xmath159 and at @xmath153 in @xmath160 . now , @xmath161 has @xmath162 s at all positions @xmath163 . furthermore , @xmath160 has a @xmath162 at position @xmath151 and shares the first @xmath151 letters with @xmath149 . schematically , we get the following structure of @xmath139 : @xmath164 here we have omitted ( @xmath165 ) all letters that are not necessary and by @xmath166 we denote the part that @xmath149 and @xmath160 share . we now check for _ pairwise coincidences _ , i.e. , for @xmath167 we check whether there is a @xmath44 and a @xmath168 such that @xmath169 . so we pick @xmath170 , @xmath171 . suppose @xmath172 ( otherwise we interchange @xmath173 and @xmath174 ) . then @xmath174 either equals @xmath157 or at least @xmath175 has an @xmath157 ( every @xmath176 , @xmath45 has one ) . in the first case take @xmath177 , otherwise @xmath178 . observe that there are @xmath151 successive @xmath162 s in @xmath161 . so , if we look at @xmath179 and @xmath180 , we get the following : on the one hand , there are @xmath151 successive @xmath162 s somewhere in @xmath180 , say at positions @xmath181 . on the other hand , in @xmath179 , there is at one of these positions @xmath181 either a @xmath162 , and we have a pairwise coincidence , or an @xmath150 . say there is an @xmath150 at @xmath182 with @xmath183 . then in @xmath184 and @xmath185 we have pairwise coincidences at positions @xmath186 ( the @xmath166 s of  ( [ eq : kol2m2_e ] ) ) .    from this pairwise coincidences we get a coincidence inductively : we start with two letters @xmath187 and after @xmath188 substitutions we have a pairwise coincidence , say at @xmath189 . now a third letter @xmath190 may have something else at @xmath191 , but whatever it is , in @xmath192 ( @xmath193 ) all three coincide somewhere at a position @xmath194 with @xmath195 . since there are @xmath196 letters , we get a coincidence after at most @xmath197 substitutions ( i.e. , there is a @xmath198 such that all @xmath199 are the same for all @xmath45 ) . the structure of @xmath138 is different . we have @xmath200 and therefore @xmath201 . let us first show an example , with @xmath202 and @xmath203 : @xmath204 since the positions of two consecutive @xmath150 s in the sequence differ by at most @xmath151 , there is an @xmath150 in every @xmath205 with @xmath45 ( note that @xmath138 is a substitution of constant length @xmath206 ) . again we look for pairwise coincidences , so choose @xmath207 . then there is ( at least ) one @xmath150 in @xmath208 , say at position @xmath189 , and ( at least ) one in @xmath209 , say at position @xmath194 . since there can be more than one @xmath150 in either , we choose @xmath210 such that @xmath211 is minimal . we further choose @xmath187 such that @xmath212 ( in the case @xmath213 , e.g. , @xmath214 and @xmath215 in the above example , we are already done ) . if we look at @xmath208 and @xmath209 , there are two cases each ( and therefore four cases , if we look at the combinations ) : either @xmath216 or @xmath217 and either @xmath218 or @xmath219 . this is all that can occur by the chosen minimality of @xmath220 . now we examine the case where @xmath221 and @xmath218 . we want to show that @xmath222 and @xmath223 have a pairwise coincidence . let us look at the @xmath150 s in @xmath224 only ( we use again the above example , but the reasons given apply for arbitrary @xmath151 , @xmath225 ) : @xmath226 first we number the @xmath150 s with @xmath227 ( left to right in @xmath224 and top ( @xmath228 ) to bottom ( @xmath229 ) ) and we will speak of the @xmath59th @xmath150 ( with @xmath230 ) according to that number . we observe the following :    * let @xmath231 . if the @xmath59th @xmath150 occurs at position @xmath232 in @xmath224 , then the @xmath233-st @xmath150 occurs at position @xmath234 in @xmath235 . if the @xmath59th @xmath150 occurs at position @xmath236 , then the @xmath233-st @xmath150 occurs at @xmath237 in the same @xmath224 . * let @xmath238 . if the @xmath59th @xmath150 occurs at position @xmath232 , then the @xmath239-st @xmath150 occurs at @xmath234 in the same @xmath224 . if the @xmath59th @xmath150 occurs at position @xmath240 in @xmath224 , then the @xmath239-st @xmath150 occurs at position @xmath237 in @xmath241 . and the position of the corresponding @xmath150 . we call this the `` contrary line break property '' . ] * the second and the @xmath242-th @xmath150 occur at the same position @xmath151 in @xmath243 , respectively @xmath244 . with the previous two observations we get : the @xmath59th and the @xmath245-th @xmath150 occur at the same position for @xmath246 . * the first and the @xmath247-st @xmath150 occur in @xmath224 where there is at least one more @xmath150 . this is obvious for the first @xmath248 , for the @xmath247-st observe that if it occurs at position @xmath249 , then there is also one at @xmath250 , and if it occurs at position @xmath251 , then there is also one at @xmath252 . these observations are based on the facts that the length of the substitution is @xmath95 and that the position of the @xmath150 s in the sequence are separated by @xmath151 or @xmath225 only . now the fact that the @xmath248 always occur in ascending order ( i.e. , we have @xmath253 and not @xmath254 or something else ) together with the first two observations essentially gives us an algorithm , which always yields a pairwise coincidence in @xmath222 and @xmath223 . let us explain it in our example  ( [ eq : ex - kol106 ] ) : suppose we have @xmath255 and @xmath256 . then we have @xmath257 and @xmath258 . the first step is always to reduce @xmath194 by one , so we have @xmath259 . we have @xmath260 , but there is a second @xmath150 in @xmath244 ( @xmath157 occurs at position @xmath261 in @xmath174 ! ) , so we can increment @xmath189 by @xmath77 and get @xmath262 . we have @xmath263 , and @xmath264 and @xmath265 both have an @xmath150 at position @xmath5 ( the third respectively the ninth @xmath150 ) . therefore we get a pairwise coincidence in @xmath266 and @xmath267 . this algorithm relies on the `` contrary line break property '' . the other three cases are mutatis mutandis the same , see the positions of the @xmath150 s in  ( [ eq : kol106 ] ) . so , starting with any two @xmath207 we get a pairwise coincidence in @xmath222 and @xmath223 . inductively like before , we get a coincidence after at most @xmath268 substitutions . therefore we have established the following . [ thm : spectrum ] @xmath125 with @xmath51 of  ( [ eq : subs1 ] ) has pure point dynamical spectrum . also , the dynamical system of the substitutions of constant length as defined implicitly in  ( [ eq : subs2 ] ) and @xmath269 of  ( [ eq : subs3 ] ) and ( [ eq : kol2m2 ] ) have pure point dynamical spectrum . we have @xmath270 where @xmath117 is the addition of @xmath118 and @xmath127 the addition of @xmath77 .    remark : let @xmath42 be a primitive substitution of constant length @xmath0 and height @xmath77 over the alphabet @xmath271 . then we can define the coincidence matrix @xmath272 , which is a quadratic @xmath273 matrix . the entries are defined as follows ( where @xmath274 and @xmath275 ) : @xmath276 note that the substitution matrix @xmath35 is a submatrix of @xmath272 , since @xmath277 . also , @xmath272 has row sums @xmath0 . with this definition , lemma  [ lem : dekiii7 ] reads as follows . * proposition x.1 ) different from  @xcite . the coincidence matrix there has dimension @xmath278 and has the form ( with the proper enumeration of the pairs ) @xmath279 while the one defined above has the form @xmath280 here @xmath281 , @xmath282 , @xmath283 are quadratic matrices and @xmath281 is the substitution matrix . the proposition is true for both matrices , the proof is analogous . ] for @xmath66 are equivalent :    1 . @xmath66 has pure point dynamical spectrum . 2 .   @xmath0 is a simple eigenvalue of the corresponding coincidence matrix @xmath272 . obviously , @xmath0 is an eigenvalue of @xmath272 ( @xmath284 is a stochastic matrix with row sum @xmath77 ) . now the above proof of theorem  [ thm : spectrum ] translates to the following statements for @xmath272 :    * for @xmath139 , the third power of the coincidence matrix , @xmath285 , has a column ( @xmath286 ) with nonzero entries only . * for @xmath138 , the square of the coincidence matrix , @xmath287 , has a column ( @xmath288 ) with nonzero entries only . * lemma x.3 ) let @xmath289 be a quadratic matrix with nonnegative integral entries and row sums @xmath0 . if @xmath290 for all @xmath291 and a fixed @xmath38 , then @xmath0 is a simple eigenvalue of @xmath289 . this establishes the desired result that @xmath0 is a simple eigenvalue for the coincidence matrix @xmath272 of @xmath138 ( @xmath139 ) , since @xmath292 ( @xmath293 ) is a simple eigenvalue of @xmath287 ( @xmath285 ) . we end this section with an example . we take @xmath139 for @xmath96 , @xmath97 and therefore the substitution @xmath294 we get the coincidence matrix @xmath295 here , already @xmath287 has columns with positive entries only . the eigenvalues of @xmath272 are @xmath296 . a _ model set _ @xmath297 ( or _ cut - and - project set _ ) in _ physical space _ @xmath298 is defined within the following general cut - and - project scheme , see  @xcite , @xmath299 where the _ internal space _ @xmath300 is a locally compact abelian group , and @xmath301 is a _ lattice _ , i.e. , a co - compact discrete subgroup of @xmath302 . the projection @xmath303 is assumed to be dense in internal space , and the projection @xmath304 into physical space has to be one - to - one on @xmath305 . the model set @xmath297 is @xmath306 where the _ window _ @xmath307 is a relatively compact set with nonempty interior . let @xmath60 be a bi - infinite sequence over @xmath308 and @xmath309 be a ( bounded ) function which assigns to every letter a complex number ( the _ scattering strength _ ) . then the _ autocorrelation coefficients _ @xmath310 are given by @xmath311 provided the limits exist . we write @xmath312 for the dirac measure at @xmath313 , i.e. , @xmath314 for @xmath76 continuous . then the _ correlation measure _ @xmath315 of @xmath60 is given by @xmath316 and the _ _ diffraction spectrum _ _ as an atomic chain , where there is an atom of type @xmath317 at position @xmath225 with scattering strength @xmath318 . we represent this atom as @xmath319 and therefore get a ( countable ) sum of weighted dirac measures with autocorrelation @xmath315 . ] is given by the fourier transform @xmath320 of this measure . if @xmath320 is a sum of dirac measures only , i.e. , @xmath321 with a countable set @xmath322 ( for any choice of complex numbers @xmath323 ) , then @xmath60 is _ pure point diffractive _ , i.e. , the diffraction spectrum consists of _ bragg peaks _ only . also , the @xmath324 s are the square of the absolute value of the corresponding _ fourier - bohr coefficient _ at @xmath59 and therefore non - negative ( real ) numbers . if there is no dirac measure in @xmath320 , except one at position @xmath9 , @xmath325 , which is determined by the density of the structure only , then the diffraction spectrum is _ continuous_.    for substitutive systems , the diffraction spectrum and the spectrum of the corresponding dynamical system are closely related , see  @xcite . [ prop : lm1 ] ( * ? ? ? * corollary 1 . ) let @xmath42 be a primitive substitution of constant length @xmath0 with height @xmath77 over @xmath271 , where @xmath60 is a fixed bi - infinite word of @xmath42 . define @xmath326 for all @xmath45 . we have @xmath327 , where @xmath328 denotes disjoint union . then the following are equivalent :    1 . @xmath42 admits a coincidence . 2 .   the @xmath329 s are model sets for @xmath330 3 . the sequence @xmath60 and the sets , where we set @xmath331 and @xmath332 for all @xmath333 . ] @xmath329 are pure point diffractive . note that properties `` one - to - one '' and `` dense '' are obvious , the interesting part is that there exist relatively compact windows ( with respect to the @xmath0-adic topology ) with nonempty ( ! ) interior . we will discuss this point in the next section .    with this proposition , we know that @xmath138 and @xmath139 ( as defined implicitly in  ( [ eq : subs3 ] ) ) generate sequences which are pure point diffractive . but we got every letter @xmath45 for the appropriate alphabet for @xmath138 , @xmath139 by building four - letter blocks in the substitution rule  ( [ eq : subs0 ] ) , e.g. , in  ( [ eq : kol42 ] ) we have @xmath334 , @xmath335 and @xmath336 . such a deterministic substitution rule ( i.e. , kol@xmath21 is _ local derivable _ from the sequence generated by @xmath138 , respectively @xmath139 ) does not change the nature of the diffraction spectrum , only the fourier - bohr coefficients . the diffraction spectrum of kol@xmath21 can be calculated from the one generated by @xmath138 , @xmath139 as follows : for kol@xmath21 we only have scattering strengths @xmath337 and @xmath338 . if we therefore choose @xmath339 ( @xmath45 with respect to @xmath138 , @xmath139 ) according to its four - letter - composition in @xmath340 , then the diffraction spectrum @xmath320 of @xmath138 , @xmath139 is also a diffraction spectrum of kol@xmath21 , where kol@xmath21 is realized as an atom chain with atoms not on @xmath341 but on @xmath342 ( we get the diffraction spectrum of kol@xmath21 realized on @xmath341 by a simple rescaling with the factor @xmath343 ) . for the example  ( [ eq : kol42 ] ) , this means that we choose @xmath344 so in this case , the diffraction spectrum of kol@xmath84 is given by the one of @xmath345 only . all kol@xmath21 are pure point diffractive .    for a sequence @xmath346 , where @xmath347 is a bi - infinite fixed point of a primitive substitution and @xmath348 is a morphism ( where @xmath60 ( @xmath347 ) is a sequence over @xmath349 ( @xmath350 ) ) , the following statements are equivalent :    1 . the dynamical system of @xmath60 has pure point dynamical spectrum . @xmath60 has pure point diffraction spectrum . this is a ( weak ) conclusion of  ( * ? ? ? * theorem 3.2 ) . we therefore obtain the following .    all kol@xmath21 have pure point diffraction and pure point dynamical spectrum . remarks : the dynamical system of kol@xmath21 is isomorphic to @xmath351 where @xmath117 is the addition of @xmath118 and @xmath127 the addition of @xmath77 . this can be seen by the fact that we get kol@xmath21 from the one generated by @xmath51 of  ( [ eq : subs1 ] ) by the substitution @xmath352 , @xmath34 , which corresponds just to doubling each letter in the latter one ( see the substitution matrices in  ( [ eq : subs1 ] ) and footnote  [ foo : note ] ) .    if we consider how to get from  ( [ eq : subs2 ] ) , respectively  ( [ eq : subs3 ] ) , to kol@xmath21 and compare this to proposition  [ prop : lm1 ] , we get ( compare with  ( [ eq : dynspec ] ) ) : kol@xmath21 , respectively @xmath353 , @xmath354 are model sets for @xmath355 where @xmath356 ( as before , @xmath357 , where @xmath124 are the distinct primes dividing @xmath95 . ) the diffraction spectrum calculated from this cut - and - project scheme is consistent with the previous one , since the fourier - bohr coefficients which arise in each @xmath358 separately are weighted by factors @xmath359 or @xmath360 which depend on the element of the cyclic group @xmath361 ( similar for the case @xmath362 ) , compare with  @xcite  but this is just how we calculated the @xmath339 s from @xmath337 and @xmath338 . let us show how one calculates the diffraction spectrum of @xmath345 of  ( [ eq : kol42 ] ) explicitly . this substitution can be written in recursive equations for @xmath363 , @xmath364 and @xmath345 by observing at which position in which substitution a certain letter occurs ( e.g. , @xmath162 occurs in @xmath152 at positions @xmath77 and @xmath5 and in @xmath160 at position @xmath5)-adic space , because multiplication by a factor of @xmath99 is a contraction in the @xmath99-adic topology . the closure of the windows in the @xmath99-adic internal space is therefore given by the unique compact solution of this ifs by ( a generalized version of ) hutchinson s theorem  ( * ? ? ? * section 3.1(3 ) ) . this method is well known for unimodular substitutions of pisot - type , see  @xcite , @xcite and the vast literature about rauzy fractals . similar results apply for all primitive substitutions of constant length @xmath0 in @xmath0-adic space . ] : @xmath365 where @xmath366 . iterating these equations and using @xmath367 , one gets : @xmath368 the fourier transform of each lattice coset @xmath369 is easy to calculate : @xmath370 every fourier - bohr coefficients of @xmath371 are then given by the sum of the fourier - bohr coefficients of the corresponding @xmath372 . the structure of @xmath345 ( similar for all @xmath329 that occur for substitutions of constant length ) as a union of a countable but infinite set of ( periodic ) lattice cosets @xmath373 gives rise to the name _ limit - periodic _ , see  @xcite . the _ support _ of the bragg peaks of kol@xmath21 is given by @xmath374 where @xmath375 are the distinct primes dividing @xmath95 and @xmath376 is the cyclic group of  ( [ eq : cyclicgroup ] ) . however , there need not be a bragg peak on every point of the support , e.g. , kol@xmath377 is equivalent to a substitution @xmath138 of constant length @xmath378 , but the positions of a letter @xmath248 , @xmath379 are separated by multiples of @xmath5 : the support in this case is better described by @xmath380 than by @xmath381 . so far , we have talked in an `` abstract '' way about the @xmath0-adic internal space . usually the discussion ends at this point , but we want to `` visualize '' this @xmath0-adic space . we hope that by doing this , we also gain some intuition for such spaces and the meaning of @xmath14-adic internal spaces for model sets . recall that a @xmath14-adic integer can be written as a formal series @xmath382 with integral coefficients @xmath383 satisfying @xmath384 ( _ hensel expansion _ ) . for the following , we identify a @xmath14-adic integer @xmath385 with the sequence @xmath386 of its coefficients . the set of all @xmath14-adic integers ( a ring ) is written as @xmath387 , while the field of @xmath14-adic numbers is written as @xmath388 and can be seen as the set of all laurent series @xmath389 with @xmath390 . there is a @xmath14-adic _ valuation _ @xmath391 defined by @xmath392 , which gives rise to the @xmath14-adic _ metric _ with @xmath393 and @xmath394 ( and @xmath388 and @xmath387 are the completions of @xmath395 and @xmath341 with respect to the @xmath14-adic metric ) . so with respect to the @xmath14-adic metric , two numbers in @xmath341 are close if their difference is divisible by a high power of @xmath14 . note that this is a _ non - archimedean _ absolute value ( i.e. , @xmath396 for all @xmath397 ) and we therefore get some `` strange '' properties : all triangles are isosceles , every point inside a ball @xmath398 is the center of this ball , all balls are open and closed , etc . , see  @xcite .    for euclidean models ( see  ( * ? ? ? * section 1.2 ) ) of @xmath387 we only need to know the formal series @xmath399 . we can even show models for @xmath120 , where we do not make use of  ( [ eq : ladic ] ) . for this , we use the addressing scheme known for fractals , for example in the sierpinsky gasket , see  ( * ? ? ? * chapter iv . ) :    = 0.65    now the interesting thing here is that each point in the sierpinsky gasket has a unique address ( at least if we do not take the usual connected sierpinsky gasket but the totally disconnected version ; this can be obtained by using a contraction factor less than @xmath400 in the ifs for the sierpinsky gasket ) . so each point in the sierpinsky gasket corresponds to a sequence @xmath386 with elements @xmath401  this is just the hensel expansion of the @xmath99-adic integers . similarly , the cantor set is such a geometric encoding of the @xmath5-adic integers . `` reasonable '' geometric representations of @xmath120 in @xmath402 are those , where the sets @xmath403 of points starting with the same address are represented by objects of the same size for a fixed @xmath404 . therefore we get that in @xmath405-dimension , @xmath120 with @xmath406 can reasonably be represented , if we do not make use of  ( [ eq : ladic ] ) . note that we can represent @xmath407 either in @xmath408 or @xmath409 . this geometric representation surely fails for some @xmath14-adic ( or @xmath0-adic ) properties ( all triangles are isosceles , every point inside a ball is its center , etc . ) , but some are also `` preserved '' : points which are close in the @xmath14-adic topology are also close in this geometric representation and the representation as totally disconnected fractal corresponds to the totally disconnected field @xmath388 , @xmath387 and its geometric models are both compact sets . and balls in the @xmath14-adic topology correspond to scaled down copies of the whole fractal . we like to conclude this section with our example from  ( [ eq : kol42 ] ) . the @xmath99-adic geometric models are given in figure  [ fig : kol42 ] . observe that , in the two - dimensional representation , the parts ( according to our above addressing scheme for the sierpinsky gasket ) @xmath410 , @xmath411 and @xmath412 are colored by only one color . this corresponds to the fact that at positions @xmath413 are @xmath150 s , on @xmath414 are @xmath415 s and on @xmath416 are @xmath162 s only in the bi - infinite sequence . so , large patches of the same color in the geometric representations correspond to lattice cosets @xmath417 with small @xmath418 . a similar addressing scheme can be used for the one - dimensional representation ( and in fact for all @xmath0-adic representations ) . = 0.5    = 0.5 it is a pleasure to thank m.  baake , r.v . moody and o.  redner for discussions and the german research council ( dfg ) for financial support . fogg , `` substitutions in dynamics , arithmetics and combinatorics '' , _ lecture notes in mathematics _ * 1784 * , edited by v.  berth , s.  ferenczi , c.  mauduit and a.  siegel , springer , 2002 ; available at .                          e.a . robinson , jr . , `` the dynamical theory of tilings and quasicrystallography '' , in _ ergodic theory of @xmath419-actions _ , edited by m.  pollicott and k.  schmidt , cambridge u.  p. , cambridge , 1996 , pp .  451473 .
we consider ( generalized ) kolakoski sequences on an alphabet with two even numbers . they can be related to a primitive substitution rule of constant length @xmath0 
 . using this connection , we prove that they have pure point dynamical and pure point diffractive spectrum , where we make use of the strong interplay between these two concepts . 
 since these sequences can then be described as model sets with @xmath0-adic internal space , we add an approach to `` visualize '' such internal spaces .