Count of primitive words

Abstract. The present paper concerns with the formula of the count of primitive words and ex-changeable primitive words.     

Note on primitive words

This note presents an example that disproves, forn=4, Weinbaum’s conjecture, that ifw is a cyclically reduced primitive word inF _n such that all the generatorsx∈X appear inw then some cyclic permutation ofw can be partitioned inton words generatingF _n :w≡uv,vu≡s ₁ s ₂…s _n , <s ₁,s ₂,…s _n >=F _n .     

Automata accepting primitive words

LeA be an automaton whose set of inputs equalsX (|X|≧2) and whose cardinality of the set of states equalsn (n≧2), and letQ be the set of all primitive words overX. ByT(A) we denote the language accepted byA. In this paper, we give the following results:

Weinbaum factorizations of primitive words

C. M. Weinbaum [1] showed the following: Let w be a primitive word and a be letter in w. Then a conjugate of w can be written as uv such that a is a prefix and a suffix of u, but v neither starts nor ends with a, and u and v have a unique position in w as cyclic factors. The latter condition means that there is exactly one conjugate of w having u as a prefix and there is exactly one conjugate of w having v as a prefix. It is this condition which makes the result non-trivial. We give a simplified proof for Weinbaum’s result. Guided by this proof we exhibit quite different, but still simple, proofs for more general statements. For this purpose we introduce the notion of Weinbaum factor and Weinbaum factorization.     

Ternary forms as invariants of Markoff forms and other SL2 (Z)-bundles

Consider the free group Γ = {A,B} generated by matrices A, B in SL₂(Z). We can construct a ternary form Φ(x,y,z) whose GL₃(Z) equivalence class is invariant, as it depends on Γ and not the choice of generators. If Γ is the commutator of SL₂(Z), then the generating matrices have fixed points corresponding to different fields and inequivalent Markoff forms, but they are all biuniquely determined by Φ = -z²+ y(2x+y+z) to within equivalence. When referred to transformations A, B of the upper half plane, this phenomenon is interpreted in terms of inequivalent homotopy elements which are primitive for the perforated torus.     

Marcus contextual languages consisting of primitive words

In this paper we prove that the language of all primitive (strongly primitive) words over a nontrivial alphabet can be generated by certain types of Marcus contextual grammars.     

Markoff triples and quasifuchsian groups

We study the global behaviour of trees of Markoff triples overthe complex numbers. We relate this to the space of type-preservingrepresentations of the punctured torus group into SL(2,C). Inparticular, we explore which Markoff triples correspond to quasifuchsianrepresentations. We derive a variation of McShane's identityfor quasifuchsian groups. In the case of non-discrete representations,we attempt to relate the asymptotic behaviour of Markoff triplesto the realisability of laminations in hyperbolic 3-space. Wealso consider how some of these issues might be related formore general surfaces. 1991 Mathematics Subject Classification:57M50.     

Generalized Markoff maps and McShane's identity

We study the (relative) SL(2,C) character varieties of the one-holed torus and the action of the mapping class group on the (relative) character variety. We show that the subset of characters satisfying two simple conditions called the Bowditch Q-conditions is open in the relative character variety and that the mapping class group acts properly discontinuously on this subset. Furthermore, this is the largest open subset for which this holds. We also show that a generalization of McShane's identity holds for all characters satisfying the Bowditch Q-conditions. Finally, we show that further variations of the McShane-Bowditch identity hold for characters which are fixed by an Anosov element of the mapping class group and which satisfy a relative version of the Bowditch Q-conditions, with applications to identities for incomplete hyperbolic structures on punctured torus bundles over the circle, and also for closed hyperbolic 3-manifolds which are obtained by hyperbolic Dehn surgery on such manifolds.     

A primitive derivation and logarithmic differential forms of Coxeter arrangements

Let W be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.     

Determination of holomorphic modular forms by primitive Fourier coefficients

We prove that Siegel modular forms of degree greater than one, integral weight and level N, with respect to a Dirichlet character of conductor are uniquely determined by their Fourier coefficients indexed by matrices whose contents run over all divisors of . The cases of other major types of holomorphic modular forms are included. The author is supported by the Grant-in-Aid for JSPS fellows.     

Continued fractions and the Markoff tree

We give here a full account of Markoff's celebrated result on badly approximable numbers. The proofs rely exclusively on the classical theory of simple continued fractions, together with Harvey Cohn's method using words in the free group with two generators for the determination of the structure of periods of the continued fractions of Markov irrationals. Appendix A gives a short self-contained presentation of the results on continued fractions used here and Appendix B gives short proofs of some results on the still open uniqueness problem for Markoff numbers.     

On some properties of the decoding of admissible words for primitive substitutions

A number of properties of decoding for admissible sequences are studied, some of which were studied earlier, and some are new. Several conditions under which a substitution possesses these properties are obtained. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 127–136. Supported by Russian Foundation for Basic Research, grant 94-01-00921. Translated by A. N. Livshits.     

Formal Markoff maps are positive

This note defines a family of Laurent polynomials indexed in which generalize the Markoff numbers and relate to the character variety of the one-cusped torus. We describe which monomials appear in each polynomial and prove all the coefficients are positive integers. We also conjecture a generalization of that positivity result.      

Subsequent to a theorem of Markoff

Let f(x, y) be an indefinite binary quadratic form, d(f) its discriminant, m(f) the infimum of |f(x, y)| over all integers x, y not both zero, and put $\text{μ(f) = m(f)d(f)}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$. In this paper we prove the existence of countably many disjoint open intervals I_j contained in $\text{0 ≤ x ≤}\text{1}\text{3}$ such that there is no f with μ(f) in I_j (j = 1, 2,…) and such that for any interval I containing two intervals I_j, I_k there is an f with μ(f) in I.     

On mean values of mollifiers and L-functions associated to primitive cusp forms

Mathematische Zeitschrift - We study the second moment of the L-function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers. This family is a...     

Häufungspunkte im Markoff Spektrum

Archiv der Mathematik -