locating factors of a characteristic word via the generalized zeckendorf representation of numbers let a be an irrational number with 0 1 for a factor w of f a the location of w is the set of all positions in f a at which w occurs the locations of all factors of f a have been determined by chuan and ho recently in this paper we obtain other formulas for the locations of factors of f a using the generalized zeckendorf representation of nonnegative integers these results are equivalent to the known results obtained by chuan and ho in the case a 3 52 we compute the longest common prefix of any two suffixes of f a and compute the order number and location index of each factor of f a given its length and a position in f a at which it begins

jet schemes of locally complete intersection canonical singularities let x be a variety defined over an algebraically closed field k of characteristic zero the mth jet scheme xm of x is a scheme whose closed points over x x are morphisms ox x k t tm 1 when x is a smooth variety this is an affine bundle over x of dimension m 1 dim x the space of arcs x of x is the projective limit x proj limm xm our main result is a proof of the following theorem which was conjectured by eisenbud and frenkel

words and forbidden factors given a finite or infinite word v we consider the set m v of minimal forbidden factors of v we show that the set m v is of fundamental importance in determining the structure of the word v in the case of a finite word w we consider two parameters that are related to the size of m w the first counts the minimal forbidden factors of w and the second gives the length of the longest minimal forbidden factor of w we derive sharp upper and lower bounds for both parameters we prove also that the second parameter is related to the minimal period of the word w we are further interested to the algorithmic point of view indeed we design linear time algorithm for the following two problems i given w construct the set m w and conversely ii given m w reconstruct the word w in the case of an infinite word x we consider the following two functions gx that counts for each n the allowed factors of x of length n and fx that counts for each n the minimal forbidden factors of x of length n we address the following general problem what information about the structure of x can be derived from the pair gx fx we prove that these two functions characterize up to the automorphism exchanging the two letters the language of factors of each single infinite sturmian word

palindromization and construction of markoff triples abstract the markoff equation is the diophantine equation x 2 y 2 z 2 3 x y z a solution is called a markoff triple we give a bijection between the free monoid on two letters and the set of markoff triples using the palindromization map of aldo de luca in our construction special christoffel words appear whose lengths are markoff numbers we study their standard and palindromic factorizations and show that they are self dual

renormalisation in implicit complex maps abstract we study the dynamics of a family of implicit complex maps restricted to an invariant circle the renormalisation analysis for golden mean rotation number is given we find a one parameter family of universality classes and relate this to a line of fixed points of the square of the circle map renormalisation transformation these period 2 points are pairs of fractional linear maps

towards the singular locus of the space of arcs we give a theory on the space of arcs x infty of a singular variety x based on the finiteness property of the stable points of x infty we also show how computations can be made in this theory as a consequence we obtain a process of determining invariants of x from its space of arcs x infty and from its spaces of wedges arcs in the space of arcs more precisely we give two concepts of regularity for the stable points of x infty and we show that we have regularity in codimension 1 and for all stable points of the space of arcs of a curve as another application we show concrete examples obtained from the spaces of arcs of noetherian 1 dimensional local rings which are analytically ramified

factors of characteristic words location and decompositions let a be an irrational number with 0 a 1 and let f a be the characteristic word of a the location of a factor w of f a is defined to be the set of all positions in f a at which w occurs in this paper an explicit formula of the location of each factor w of f a is obtained the decompositions red a w and sod a w of f a associated with w are established the former one involves all return words of w while the latter one involves all occurrences of w and the corresponding separate factors and overlap factors these results generalize the work by the present authors 2005 z x wen and z y wen 1994 g melancon 1999 w t cao and z y wen 2003 i m araujo and v bruyere 2005 and a glen 2006

fine and wilf s theorem for three periods and a generalization of sturmian words we extend the theorem of fine and wilf to words having three periods we then define the set 3 per of words of maximal length for which such result does not apply we prove that the set 3 per and the sequences of complexity 2n 1 introduced by arnoux and rauzy to generalize sturmian words have the same set of factors

renormalizations and rigidity theory for circle homeomorphisms with singularities of the break type circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion we establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result namely we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same size of a break

on the number of factors of sturmian words abstract we prove that for m 1 card a m 1 m i 1 m i 1 i where a m is the set of factors of length m of all the sturmian words and is the euler function this result was conjectured by dulucq and gouyou beauchamps 1987 who proved that this result implies that the language m 0 a m c is inherently ambiguous we also give a combinatorial version of the riemann hypothesis

studies on finite sturmian words several properties of finite sturmian words are proved the inverse of the richomme seebold bijection between factor sets of given length of sturmian sequences and left special sturmian words is constructed using circular factors of christoffel words and studied the labbe bijection between christoffel words and left special sturmian words is introduced and studied factor sets are classified by two types the two classes are characterized by conjugation and periodicity properties the two classes are related to the rauzy graph a normal form for finite sturmian words is given following a result of de luca and de luca several classes of sturmian words are characterized through this normal form left or right special words bispecial words palindromes central words the normal form is used to derive a proof of the lipatov mignosi formula for the number of sturmian words of given length to produce a linear algorithm which checks if a word is sturmian and which computes its normal form to construct algorithmically the contraction and completion in various classes of sturmian words to compute the de luca palindromic closure of any sturmian word and prove that it is sturmian the completion is related to the de luca palindromization function and it is shown that the sets of directive words of central words of even and odd length are rational group languages christoffel words are characterized in several ways an extension of a result of chuan an extension of a result of droubay pirillo by counting circular factors by counting lyndon factors a special christoffel word is constructed having length a given markoff number a bivariate count of special sturmian words is given using ideas of bedaride domenjoud jamet and remy open problems are given

minimal forbidden factors of circular words minimal forbidden factors are a useful tool for investigating properties of words and languages two factorial languages are distinct if and only if they have different antifactorial sets of minimal forbidden factors there exist algorithms for computing the minimal forbidden factors of a word as well as of a regular factorial language conversely crochemore et al ipl 1998 gave an algorithm that given the trie recognizing a finite antifactorial language m computes a dfa of the language having m as set of minimal forbidden factors in the same paper they showed that the obtained dfa is minimal if the input trie recognizes the minimal forbidden factors of a single word we generalize this result to the case of a circular word we also discuss combinatorial properties of the minimal forbidden factors of a circular word finally we characterize the minimal forbidden factors of the circular fibonacci words

standard sturmiam morphisms abstract we give a characterization of morphisms which preserve finite and infinite standard sturmian words the class of such morphisms coincides with the monoid d e of the endomorphisms of a where a a b generated by the two elementary morphisms e which interchanges the letter a with b and d which is the fibonacci morphism defined as d a ab d b a some new properties of these morphisms are shown in particular we prove the following result let x n n 0 be the sequence of sets inductively defined as x 0 e x 1 a x n 1 ax n for n 1 where e denotes the empty word and is the operator of left palindrome closure associating to each word w the shortest word having w as suffix for all n 0 one has x n 1 e g d x n a where denotes the identity map from this we derive a new characterization of the set per of all words w having two periods p and q which are coprimes and such that w p q 2 indeed one has that per aa xcup n 0 e l od x n a

the attractor of renormalization and rigidity of towers of critical circle maps we demonstrate the existence of a global attractor with a cantor set structure for the renormalization of critical circle mappings the set is invariant under a generalized renormalization transformation whose action on is conjugate to the two sided shift with a countable alphabet

existence and properties of p tupling fixed points we prove the existence of fixed points of p tupling renormalization operators for interval and circle mappings having a critical point of arbitrary real degree r 1 some properties of the resulting maps are studied analyticity univalence behavior as r tends to infinity

repetitions in sturmian strings in this paper we apply a simple representation of sturmian strings which we call a reduction sequence to three algorithms the first algorithm accepts as input a given finite string x and determines in time whether or not x is sturmian the second algorithm is a modification of the first that in the case that x is sturmian outputs a reduction sequence for a superstring u of x that is a prefix of an infinite sturmian string the third algorithm uses the reduction sequence of u to compute all the repetitions in u in time thus extending a recent result for fibonacci strings the third algorithm is also based on a characterization of the repetitions in a sturmian string that describes them compactly in terms of runs finally for every integer r 4 we show how to construct an infinite sturmian string that contains maximal repetitions of exponents 2 3 r 1 but none of exponent r

infinite words with linear subword complexity abstract in section 1 we study the relations among some combinatorial properties of infinite words especially in the case of infinite words with linear subword complexity the main result of section 1 concerns a permutation property of infinite words with linear complexity in section 2 we investigate the special case of the sturmian infinite words the main result is that a sturmian infinite word associated to a real number contains no k th powers iff has a continued fraction expansion with bounded partial quotients in section 3 it is shown that some of the results of sections 1 and 2 are optimal

words and special factors in this paper we consider sets of factors of a given finite word over a finite alphabet which permit us to reconstruct the entire word this analysis is based on the notion of special factor a factor u of a finite word w is called right resp left special if there exist two distinct letters x and y such that ux uy resp xu yu are factors of w a factor is bispecial if it is right and left special a proper box of w is any factor of w of the kind asb with a b letters and s a bispecial factor of w the initial resp terminal box of w is the shortest prefix resp suffix of w which is an unrepeated factor a box is called maximal if it is not a proper factor of another box the main result of the paper is the following theorem maximal box theorem any finite word w is uniquely determined by the initial box the terminal box and the set of maximal boxes a consequence is that a finite word w is uniquely determined by the knowledge of its factors up to the length n max rw kw 1 where kw is the length of the terminal box and rw is the minimal natural number for which there is no right special factor of length rw some structural properties of boxes are studied another important combinatorial notion is that of superbox a superbox is any factor of w of the kind asb with a b letters and such that s is a repeated factor whereas as and sb are unrepeated factors a theorem for superboxes similar to the maximal box theorem is proved some algorithms allowing us to construct boxes and superboxes and conversely to reconstruct the word are given in this combinatorial frame we give an upper and a lower bound to the number of states of a minimal deterministic automaton recognizing the set of the factors of w these bounds are sharper than the known bounds

sturmian words structure combinatorics and their arithmetics abstract we prove some new results concerning the structure the combinatorics and the arithmetics of the set per of all the words w having two periods p and q p q which are coprimes and such that w p q 2 a basic theorem relating per with the set of finite standard sturmian words was proved in de luca and mignosi 1994 the main result of this paper is the following simple inductive definition of per the empty word belongs to per if w is an already constructed word of per then also aw and bw belong to per where denotes the operator of palindrome left closure i e it associates to each word u the smallest palindrome word u having u as a suffix we show that by this result one can construct in a simple way all finite and infinite standard sturmian words we prove also that up to the automorphism which interchanges the letter a with the letter b any element of per can be codified by the irreducible fraction p q this allows us to construct for any n 0 a natural bijection that we call farey correspondence of the set of the farey series of order n 1 and the set of special elements of length n of the set st of all finite sturmian words finally we introduce the concepts of farey tree and farey monoid this latter is obtained by defining a suitable product operation on the developments in continued fractions of the set of all irreducible fractions p q

some combinatorial properties of sturmian words abstract in this paper we give a characterization of finite sturmian words by palindrome words which generalizes a property of the fibonacci words we prove that the set st of finite sturmian words coincides with the set of the factors of all the words w such that w ab cxy with a b c palindromes x y a b and x y moreover using this result we prove that st is equal to the set of the factors of all words w having two periods p and q which are coprimes and such that w p q 2 several other combinatorial properties concerning special and bispecial elements of st are shown as a consequence we give a new and purely combinatorial proof of the enumeration formula of st

combinatorial properties of sequences defined by the billiard in paved triangles resume nous etudions certaines proprietes combinatoires de la dynamique symbolique du billard dans le triangle rectangle isocele et le demi triangle equilateral dans le cas du triangle rectangle isocele nous developpons un algorithme de construction des suites obtenues comme codage et nous calculons la complexite de ces suites lorsqu elles ne sont pas periodiques nous donnons des regles de construction des mots infinis qui codent les trajectoires de billard dans le demi triangle equilateral ceci nous a conduit a etablir un algorithme de construction du codage d un point sous une rotation munie d une partition en intervalles

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

terminal valuations and the nash problem let x be an algebraic variety of characteristic zero terminal valuations are defined in the sense of the minimal model program as those valuations given by the exceptional divisors on a minimal model over x we prove that every terminal valuation over x is in the image of the nash map and thus it corresponds to a maximal family of arcs through the singular locus of x in dimension two this result gives a new proof of the theorem of fernandez de bobadilla and pe pereira stating that for surfaces the nash map is a bijection

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

new proofs of the existence of the feigenbaum functions a new proof of the existence of analytic unimodal solutions of the cvitanovic feigenbaum functional equation g x g g x g x 1 const x r at 0 valid for all in 0 1 is given and the existence of the eckmann wittwer functions 8 is recovered the method also provides the existence of solutions for certain given values ofr and in particular forr 2 a proof requiring no computer

pseudopalindrome closure operators in free monoids we consider involutory antimorphisms of a free monoid a and their fixed points called palindromes or pseudopalindromes a palindrome reduces to a usual palindrome when is the reversal operator for any word w a the right resp left palindrome closure of w is the shortest palindrome having w as a prefix resp suffix we prove some results relating palindrome closure operators with periodicity and conjugacy and derive some interesting closure properties for the languages of finite sturmian and episturmian words in particular a finite word w is sturmian if and only if both its palindromic closures are so moreover in such a case both the palindromic closures of w share the same minimal period of w a new characterization of finite sturmian words follows in terms of periodicity and special factors of their palindromic closures some weaker results can be extended to the episturmian case by using the right palindrome closure we extend the construction of standard episturmian words via directive words in this way one obtains a family of infinite words called standard words which are morphic images of episturmian words as well as a wider family of infinite words including the thue morse word on two symbols

on the existence of fixed points of the composition operator for circle maps in the theory of circle maps with golden ratio rotation number formulated by feigenbaum kadanoff and shenker fks and by ostlund rand sethna and siggia orss a central role is played by fixed points of a certain composition operator in map space we define a common setting for the problem of proving the existence of these fixed points and of those occurring in the theory of maps of the interval we give a proof of the existence of the fixed points for a wide range of the parameters on which they depend

feigenbaum theory for unimodal maps with asymmetric critical point rigorous results we apply the methods of h epstein to prove the existence of a line of period 2 solutions of the feigenbaum period doubling renormalisation transformation these solutions govern the universal behaviour of families of unimodal maps with asymmetric critical points of degree d for which the d th derivative has differing left and right limits

the nash problem on arc families of singularities nash 21 proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution he asked if the converse also holds does every such exceptional divisor correspond to an arc family we prove that the converse holds for toric singularities but fails in general

codes of central sturmian words a central sturmian word or simply central word is a word having two coprime periods p and q and length equal to p q 2 we consider sets of central words which are codes some general properties of central codes are shown in particular we prove that a non trivial maximal central code is infinite moreover it is not maximal as a code a central code is called prefix central code if it is a prefix code we prove that a central code is a prefix resp maximal prefix central code if and only if the set of its generating words is a prefix resp maximal prefix code a suitable arithmetization of the theory is obtained by considering the bijection called ratio of periods from the set of all central words to the set of all positive irreducible fractions defined as e 1 1 and w p q resp w q p if w begins with the letter a resp letter b p is the minimal period of w and q w p 2 we prove that a central code x is prefix resp maximal prefix if and only if x is an independent resp independent and full set of fractions finally two interesting classes of prefix central codes are considered one is the class of farey codes which are naturally associated with the farey series we prove that farey codes are maximal prefix central codes the other is given by uniform central codes a noteworthy property related to the number of occurrences of the letter a in the words of a maximal uniform central code is proved

factors of characteristic words of irrational numbers let be an irrational number between 0 and 1 the characteristic word f of is defined to be the infinite word over 0 1 whose nth letter is n 1 n n 1 it is well known that for each m 1 f has exactly m 1 distinct factors of length m in this paper we shall develop a method to construct these factors under our construction the 1 sets of these m 1 factors x0 m x1 m xm m are determined these factors are increasing in the lexicographic order and their moments m x0 m m x1 m m xm m form an increasing sequence of m 1 consecutive integers some known results about generating factors of f using the unbordered words and their conjugates turn out to be consequences of our main theorem

standard sturmian words and automata minimization algorithms the study of some close connections between the combinatorial properties of words and the performance of the automata minimization process constitutes the main focus of this paper these relationships have been in fact the basis of the study of the tightness and the extremal cases of hopcroft s algorithm that is up to now the most efficient minimization method for deterministic finite state automata recently increasing attention has been paid to another minimization method that unlike the approach proposed by hopcroft is not based on refinement of the set of states of the automaton but on automata operations such as determinization and reverse and is also applicable to non deterministic finite automata however even for deterministic automata it was proved that the method incurs almost surely in an explosion of the number of the states very recently some polynomial variants of brzozowski s method have been introduced in this paper by using some combinatorial properties of words we analyze the performance of one of such algorithms when applied to a particular infinite family of automata called standard word automata constructed by using standard sturmian words in particular n log n is the worst case time complexity when the algorithm is applied to the infinite family of word automata associated to fibonacci words representing also the worst case of hopcroft s minimization algorithm

renormalization horseshoe for critical circle maps we prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one dimensional unstable direction this extends our earlier result on hyperbolicity of periodic orbits of the renormalization and completes the proof of lanford s program

germs of arcs on singular algebraic varieties and motivic integration we study the scheme of formal arcs on a singular algebraic variety and its images under truncations we prove a rationality result for the poincare series of these images which is an analogue of the rationality of the poincare series associated to p adic points on a p adic variety the main tools which are used are semi algebraic geometry in spaces of power series and motivic integration a notion introduced by m kontsevich in particular we develop the theory of motivic integration for semi algebraic sets of formal arcs on singular algebraic varieties we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of oesterle

a new geometric approach to sturmian words we introduce a new geometric approach to sturmian words by means of a mapping that associates certain lines in the nxn grid and sets of finite sturmian words of length n using this mapping we give new proofs of the formulas enumerating the finite sturmian words and the palindromic finite sturmian words of a given length we also give a new proof for the well known result that a factor of a sturmian word has precisely two return words

sturmian words lyndon words and trees abstract we prove some new combinatorial properties of the set per of all words w having two periods p and q which are coprimes and such that w p q 2 4 3 we show that aperb a b st lynd where st is the set of the finite factors of all infinite sturmian words and lynd is the set of the lyndon words on the alphabet a b it is also shown that aperb a b cp where cp is the set of christoffel primitive words such words can be defined in terms of the slope of the words and of their prefixes 1 from this result one can derive in a different way by using a theorem of borel and laubie that the elements of the set aperb are lyndon words we prove the following correspondence between the ratio p q of the periods p q p q of w per a a b and the slope w b 1 w a 1 of the corresponding christoffel primitive word awb if p q has the development in continued fractions 0 h 1 h n 1 h n 1 then has the development in continued fractions 0 h n h 2 h 1 1 this and other related results can be also derived by means of a theorem which relates the developments in continued fractions of the stern brocot and the raney numbers of a node in a complete binary tree however one needs some further results more precisely we label the binary tree with standard pairs standard tree christoffel pairs christoffel tree and the elements of per farey tree the main theorem is the following if the node w is labeled by the standard pair u v by the christoffel pair x y and by w per then uv wab xy awb the stern brocot number sb w is equal to the slope of the standard word uv and of the christoffel word xy while the raney number ra w is equal to the ratio of the minimal period of wa and the minimal period of wb some further auxiliary results are also derived

some characterizations of finite sturmian words in this paper we give some new characterizations of words which are finite factors of sturmian words an enumeration formula for primitive finite sturmian words is given moreover we provide two linear time algorithms to recognize whether a finite word is sturmian

singularities with the highest mather minimal log discrepancy this paper characterizes singularities with mather minimal log discrepancies in the highest unit interval i e the interval between d 1 and d where d is the dimension of the scheme the class of these singularities coincides with one of the classes of 1 compound du val singularities 2 normal crossing double singularities 3 pinch points and 4 pairs of non singular varieties and boundaries with multiplicities less than or equal to 1 at the point as a corollary we also obtain one implication of an equivalence conjectured by shokurov for the usual minimal log discrepancies

reprint of palindromization and construction of markoff triples abstract the markoff equation is the diophantine equation x 2 y 2 z 2 3 x y z a solution is called a markoff triple we give a bijection between the free monoid on two letters and the set of markoff triples using the palindromization map of aldo de luca in our construction special christoffel words appear whose lengths are markoff numbers we study their standard and palindromic factorizations and show that they are self dual

a periodicity result of partial words with one hole abstract the study of the combinatorial properties of strings of symbols from a finite alphabet also referred to as words is profoundly connected to numerous fields such as biology computer science mathematics and physics partial words have long been known they have been around in information and coding theory as well as molecular biology while a word can be described by a total function a partial word can be described by a partial function more precisely a partial word of length n over a finite alphabet a is a partial function from 0 n 1 into a elements of n 1 without an image are called holes a word is just a partial word without holes the notion of period of a word is central in combinatorics on words in the case of partial words there are two notions one is that of period and the other is that of local period there are many fundamental results on periods of words among them is the well known periodicity result of fine and wilf which intuitively determines how far two periodic events have to match in order to guarantee a common period this result was extended to partial words with one hole by berstel and boasson we can define the set per o of all words of maximal length for which fine and wilf s result does not apply similarly we can define the set per 1 of all partial words with one hole of maximal length for which berstel and boasson s result does not apply it is well known that per o contains a unique word up to a renaming that is binary in this note we extend this result to per 1

weak repetitions in sturmian strings in this paper we analyze the weak repetitions in sturmian strings and show that an optimally efficient algorithm to compute the weak repetitions in sturmian strings according to the output encodings defined in the literature is quadratic in the string length finally we present an encoding that leads to a linear time algorithm

on the number of episturmian palindromes episturmian words are a suitable generalization to arbitrary alphabets of sturmian words in this paper we are interested in the problem of enumerating the palindromes in all episturmian words over a k letter alphabet a k we give a formula for the map g k giving for any n the number of all palindromes of length n in all episturmian words over a k this formula extends to k 2 a similar result obtained for k 2 by the second and third authors in 2006 the map g k is expressed in terms of the map p k counting for each n the palindromic prefixes of all standard episturmian words epicentral words for any n 0 p 2 n f n 2 where f is the totient euler function the map p k plays an essential role also in the enumeration formula for the map l k counting for each n the finite episturmian words over a k similarly to euler s function the behavior of p k is quite irregular the first values of p k and of the related maps g k and l k for 3 k 6 have been calculated and reported in the paper some properties of p k are shown in particular broad upper and lower bounds for p k as well as for m 0 np k m and g k are determined finally some conjectures concerning the map p k are formulated

scaling properties and renormalization invariants for the homoclinic quasiperiodicity abstract some bifurcation scenarios near the gluing homoclinic bifurcation are considered the attracting trajectories related to irrational rotation numbers do not look like motions on a 2 torus however qualitatively they have much in common the unusual metric properties of bifurcation sequences are studied with the help of renormalization which shows that the prevalence of superexponential scaling both in the subcritical pre chaotic and supercritical parameter domains is due to the existence of singular fixed points of the corresponding renormalization transformations an approximate analysis near these points yields quantitative characteristics of scaling laws on the boundary between subcritical and supercritical regions the scaling is exponential the transformations however possess an additional invariant which characterizes the mismatch of two branches of the return map near the saddle point of the initial continuous system and influences both local for a given rotation number and global quantitative characteristics of the bifurcation diagram

coordinates at stable points of the space of arcs abstract let x be a variety over a field k and let x be its space of arcs let p e be the stable point of x defined by a divisorial valuation e on x assuming char k 0 if x is smooth at the center of p e we make a study of the graded algebra associated to e and define a finite set whose elements generate a localization of the graded algebra modulo etale covering this provides an explicit description of a minimal system of generators of the local ring o x p e if x is singular we obtain generators of p e p e 2 and conclude that embdim o x red p e embdim o x p e k e 1 where k e is the mather discrepancy of x with respect to e this provides algebraic tools for explicit computations of the local rings o x p e

graph connectivity partial words and a theorem of fine and wilf the problem of computing periods in words or finite sequences of symbols from a finite alphabet has important applications in several areas including data compression string searching and pattern matching algorithms the notion of period of a word is central in combinatorics on words there are many fundamental results on periods of words among them is the well known and basic periodicity result of fine and wilf which intuitively determines how far two periodic events have to match in order to guarantee a common period more precisely any word with length at least p q gcd p q having periods p and q has also period the greatest common divisor of p and q gcd p q moreover the bound p q gcd p q is optimal since counterexamples can be provided for words of smaller length partial words or finite sequences that may contain a number of do not know symbols or holes appear in natural ways in several areas of current interest such as molecular biology data communication dna computing etc any long enough partial word with h holes and having periods p q has also period gcd p q in this paper we give closed formulas for the optimal bounds l h p q in the case where p 2 and also in the case where q is large in addition we give upper bounds when q is small and h 3 4 5 6 or 7 no closed formulas for l h p q were known except for the cases where h 0 1 or 2 our proofs are based on connectivity in graphs associated with partial words a world wide web server interface has been established at www uncg edu mat research finewilf3 for automated use of a program which given a number of holes h and two periods p and q computes the optimal bound l h p q and an optimal word for that bound a partial word u with h holes of length l h p q 1 is optimal if p and q are periods of u but gcd p q is not a period of u

universal properties of maps of the circle with ɛ singularities following the work of collet eckmann and lanford on the feigenbaum conjecture we study the structure of the renormalization transformation introduced in 12 upon maps of the circle with critical points of the formx x

on a generalization of christoffel words epichristoffel words sturmian sequences are well known as the ones having minimal complexity over a 2 letter alphabet they are also the balanced sequences over a 2 letter alphabet and the sequences describing discrete lines they are famous and have been extensively studied since the 18th century one of the extensions of these sequences over a k letter alphabet with k 3 is the episturmian sequences which generalizes a construction of sturmian sequences using the palindromic closure operation there exists a finite version of the sturmian sequences called the christoffel words they have been known since the works of christoffel and have interested many mathematicians in this paper we introduce a generalization of christoffel words for an alphabet with 3 letters or more using the episturmian morphisms we call them the epichristoffel words we define this new class of finite words and show how some of the properties of the christoffel words can be generalized naturally or not for this class

the nash problem for surfaces we prove that nash mapping is bijective for any surface dened over an algebraically closed eld of characteristic 0

repetitive perhaps but certainly not boring in this paper some of the work done on repetitions in strings is surveyed especially that of an algorithmic nature several open problems are described and conjectures formulated about some of them

occurrences of palindromes in characteristic sturmian words this paper is concerned with palindromes occurring in characteristic sturmian words c of slope where 0 1 is an irrational as c is a uniformly recurrent infinite word any palindromic factor of c occurs infinitely many times in c with bounded gaps our aim is to completely describe where palindromes occur in c in particular given any palindromic factor u of c we shall establish a decomposition of c with respect to the occurrences of u such a decomposition shows precisely where u occurs in c and this is directly related to the continued fraction expansion of

on christoffel and standard words and their derivatives we introduce and study natural derivatives for christoffel and standard words as well as for characteristic sturmian words these derivatives which are defined as inverse images under suitable morphisms preserve the aforementioned classes of words in the case of christoffel words the morphisms involved map a to a k 1 b resp a b k and b to a k b resp a b k 1 for a suitable k 0 as long as derivatives are not just a single letter higher order derivatives are naturally obtained we define the depth of a christoffel or of a standard word as the smallest order for which the derivative is a single letter we give several combinatorial and arithmetic descriptions of the depth and tight lower and upper bounds for it

inversion of adjunction for local complete intersection varieties we extend inversion of adjunction and semicontinity of minimal log discrepancies from the case of a smooth ambient variety to the case of local complete intersection varieties

semiperiodic words and root conjugacy a factor u of a word w is called right special if there exist two distinct letters a and b such that both ua and ub are factors of w left special factors are defined symmetrically by rw resp lw we denote the minimal natural number such that there is no right resp left special factor of w of length rw resp lw moreover hw resp kw denotes the length of the shortest prefix resp suffix which cannot be extended on the left resp right in w the parameters rw lw hw and kw give interesting information on the structure of the word w we consider the class of all finite words w such that rw hw these words are called semiperiodic any periodic word is semiperiodic whereas the converse is not generally true several characterizations of semiperiodic words can be given in particular a word w is semiperiodic if and only if it has a period p w rw a further characterization of semiperiodic words relates with their infinite extensions from this characterization one derives the following result deeply related to the theorem of fine and wilf if w is a semiperiodic word having two periods p q w rw then also d gcd p q is a period of w the root rw of a word w is its prefix whose length is equal to the minimal period of w two words u and v are root conjugate if their roots ru and rv are conjugate one of the main results of the paper is the following let w be a semiperiodic word a word v has the same set of factors of length 1 rw of w if and only if v is semiperiodic and root conjugate with w some applications and extensions of this result are proved

unbordered factors of the characteristic sequences of irrational numbers abstract a finite word w over an alphabet is said to be bordered if there is a word which is both a proper prefix and a proper suffix of w in this paper the unbordered factors of the characteristic sequences f of irrational numbers between 0 and 1 are determined by definition f is the infinite word over the alphabet lcub 0 1 rcub whose nth letter is n l n n 1 it is proved that the unbordered factors of f are precisely the lyndon words over lcub 0 1 rcub and their mirror images as a biproduct a characterization of words is obtained

on functional equations associated with the renormalization of non commuting circle mappings we investigate the functional equations associated with critical non commuting circle homeomorphisms using epstein s herglotz function methods we show that there are analytic solutions of a class of functional equations closely related to period two points of the circle map renormalization transformations for rotation number with continued fraction

on the factors of the sturmian sequences resume cet article a pour objet l etude d une construction associant a toute droite de pente p q p et q premiers entre eux et q n un mot de longueur n sur l alphabet 0 1 nous montrons que nous obtenons par cette construction le langage constitue de tous les facteurs des suites de sturm nous formulons apres avoir obtenu une equation fonctionelle dont la solution est la serie generatrice de ce langage une conjecture reliant cette serie generatrice a la fonction d euler

enumeration and structure of trapezoidal words trapezoidal words are words having at most n 1 distinct factors of length n for every n 0 they therefore encompass finite sturmian words we give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration we then separate trapezoidal words into two disjoint classes open and closed a trapezoidal word is closed if it has a factor that occurs only as a prefix and as a suffix otherwise it is open we investigate open and closed trapezoidal words in relation with their special factors we prove that sturmian palindromes are closed trapezoidal words and that a closed trapezoidal word is a sturmian palindrome if and only if its longest repeated prefix is a palindrome we also define a new class of words semicentral words and show that they are characterized by the property that they can be written as uxyu for a central word u and two different letters x y finally we investigate the prefixes of the fibonacci word with respect to the property of being open or closed trapezoidal words and show that the sequence of open and closed prefixes of the fibonacci word follows the fibonacci sequence

palindromes and sturmian words abstract an infinite word x over the alphabet a is sturmian if and only if g x n n 1 for any integer n where g x n is the number of distinct words of length n occurring in x a palindrome is a word that can be read indistinctly from left to right or from right to left we prove that x is sturmian if and only if h x n 1 n mod 2 for any integer n where h x n is the number of palindromes of length n occurring in x an infinite word x over the alphabet a is generated by a morphism f if there exists a letter c a such that lim n f n c x we prove the existence of a morphism that generates the palindromes of any infinite sturmian word generated by a morphism

minimal non convex words using a combinatorial characterization of digital convexity based on words one defines the language of convex words the complement of this language forms an ideal whose minimal elements with respect to the factorial ordering appear to have a particular combinatorial structure very close to the christoffel words in this paper those words are completely characterized as those of the form uw kv where k 1 email protected v and u v w are christoffel words also by considering the most balanced among the unbalanced words we obtain a second characterization for a special class of minimal non convex words that are of the form u 2v 2 corresponding to the case k 1 in the previous form

on graphs of central episturmian words episturmian sequences are a natural extension of sturmian sequences to the case of finite alphabets of arbitrary cardinality in this paper we are interested in central episturmian words or simply epicentral words i e the palindromic prefixes of standard episturmian sequences an epicentral word admits a variety of faithful representations including as a directive word as a certain type of period vector as a parikh vector as a certain type of fine and wilf extremal word as a suitable modular matrix and as a labeled graph various interconnections between the different representations of an epicentral word are analyzed in particular we investigate the structure of the graphs of epicentral words proving some curious and surprising properties

sturmian words and the stern sequence central standard and christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart to characteristic sturmian words a natural arithmetization of the theory is obtained by representing central and christoffel words by irreducible fractions labeling respectively two binary trees the raney or calkin wilf tree and the stern brocot tree the sequence of denominators of the fractions in the raney tree is the famous stern diatomic numerical sequence an interpretation of the terms s n of stern s sequence as lengths of christoffel words when n is odd and as minimal periods of central words when n is even allows one to interpret several results on christoffel and central words in terms of stern s sequence and conversely to obtain new insight into the combinatorics of christoffel and central words one of our main results is a non commutative version of the alternating bit sets theorem by calkin and wilf we also study the length distribution of christoffel words corresponding to nodes of equal height in the tree obtaining some interesting bounds and inequalities

sturmian morphisms and a words abstract in this paper we prove that the set w of words over 0 1 coincides with the set of images of 0 under sturmian morphisms this gives an alternative approach of generating words for each w w a sturmian solution h of the equation w h 0 which is a composition of sturmian morphisms of special forms can be constructed explicitly in order to demonstrate how results on sturmian morphic images are reformulated as results on words and vice versa we give a sturmian proof for the conjugation and reversion theorem of words obtained by the present author in an earlier paper

characterisations of balanced words via orderings three new characterisations of balanced words are presented each of these characterisations is based on the ordering of a shift orbit either lexicographically or with respect to the norm 1 which counts the number of occurrences of the symbol 1

on the combinatorics of finite words in this paper we consider a combinatorial method for the analysis of finite words recently introduced in colosimo and de luca special factors in biological strings preprint 97 42 dipt matematica univ di roma for the study of biological macromolecules the method is based on the analysis of right special factors of a given word a factor u of a word w is special if there exist at least two occurrences of the factor u in w followed on the right by two distinct letters we show that in the combinatorics of finite words two parameters play an essential role the first denoted by r represents the minimal integer such that there do not exist special factors of w of length r the second that we denote by k is the minimal length of a factor of w which cannot be extended on the right in a factor of w some new results are proved in particular a new characterization in terms of special factors and of r and k is given for the set per of all words w having two periods p and q which are coprimes and such that w p q 2

characteristic sturmian words are extremal for the critical factorization theorem we prove that characteristic sturmian words are extremal for the critical factorization theorem cft in the following sense if p x n denotes the local period of an infinite word x at point n we prove that x is a characteristic sturmian word if and only if p x n is smaller than or equal to n 1 for all n 1 and it is equal to n 1 for infinitely many integers n this result is extremal with respect to the cft since a consequence of the cft is that for any infinite recurrent word x either the function p x is bounded and in such a case x is periodic or p x n n 1 for infinitely many integers n as a byproduct of the techniques used in the paper we extend a result of harju and nowotka 2002 in 18 stating that any finite fibonacci word f n n 5 has only one critical point indeed we determine the exact number of critical points in any finite standard sturmian word