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53
Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions
"... Abstract. This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2, Z) with respect to the standard fundamental domain F = {z = x+iy: −1 1 2 ≤ x ≤ ..."
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Abstract. This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2, Z) with respect to the standard fundamental domain F = {z = x+iy: −1 1 2 ≤ x ≤
SYMBOLIC DYNAMICS FOR THE MODULAR SURFACE AND BEYOND
, 2007
"... In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording ..."
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In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.
Doppler resilient Golay complementary pairs for radar,” presented at the
 IEEE Statist. Signal Process. Workshop (SSP
, 2007
"... We present a systematic way of constructing a Doppler resilient sequence of Golay complementary waveforms for radar, for which the composite ambiguity function maintains ideal shape at small Doppler shifts. The idea is to determine a sequence of Golay pairs that annihilates the loworder terms of th ..."
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We present a systematic way of constructing a Doppler resilient sequence of Golay complementary waveforms for radar, for which the composite ambiguity function maintains ideal shape at small Doppler shifts. The idea is to determine a sequence of Golay pairs that annihilates the loworder terms of the Taylor expansion of the composite ambiguity function. The ProuhetThueMorse sequence plays a key role in the construction of Doppler resilient sequences of Golay pairs. We extend this construction to multiple dimensions. In particular, we consider radar polarimetry, where the dimensions are realized by two orthogonal polarizations. We determine a sequence of twobytwo Alamouti matrices, where the entries involve Golay pairs and for which the matrixvalued composite ambiguity function vanishes at small Doppler shifts. 1.
A short proof of the transcendence of the Thue– Morse continued fraction
"... The ThueMorse sequence t = (tn)n≥0 on the alphabet {a, b} is defined as follows: tn = a (respectively, tn = b) if the sum of binary digits of n is even (respectively, odd). This famous binary sequence was first introduced by A. Thue [12] in 1912. It was considered nine years later by M. Morse [7] i ..."
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The ThueMorse sequence t = (tn)n≥0 on the alphabet {a, b} is defined as follows: tn = a (respectively, tn = b) if the sum of binary digits of n is even (respectively, odd). This famous binary sequence was first introduced by A. Thue [12] in 1912. It was considered nine years later by M. Morse [7] in a totally different context. These pioneering papers have led to a number of investigations and a broad literature devoted to t. There are many other ways to define the ThueMorse sequence. Each of them gives rise to specific interests, problems, and most of the time solutions. Such ubiquity is well described in the survey [1], where the occurrence of t in combinatorics, number theory, differential geometry, theoretical computer science, physics, and even chess is documented. For a and b distinct integers K. Mahler [6] (see also [2]) established that the sum of the series ∑ −n n≥0 tn2 is transcendental. The present note adresses another Diophantine result related to the ThueMorse sequence. It is widely believed that the continued fraction expansion of every irrational algebraic number α either is eventually periodic (and we know from Lagange’s theorem that this is the case if and only if α is a quadratic irrational) or contains arbitrarily large partial quotients. Apparently, this challenging question was first considered by A. Ya. Khintchin in [4] (see also [5], [11], or [13] for surveys or books including discussions of this subject). A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions with bounded partial quotients. In this direction, M. Queffélec [8] showed in 1998 that the ThueMorse continued fractions are transcendental. Theorem 1 (Queffélec). If a and b are distinct positive integers and t = (tn)n≥0 is the ThueMorse sequence on the alphabet {a, b}, then the number is transcendental. 1 ξ = [t0, t1, t2,...] = t0 +
Rarified sums of the ThueMorse sequence
 Trans. Amer. Math. Soc
"... Abstract. Let q be an odd number and Sq,0(n) the difference between the number of k < n, k ≡ 0mod q, with an even binary digit sum and the corresponding number of k 0 for all n. Inthispaper it is proved that ..."
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Abstract. Let q be an odd number and Sq,0(n) the difference between the number of k < n, k ≡ 0mod q, with an even binary digit sum and the corresponding number of k<n,k≡0mod q, with an odd binary digit sum. A remarkable theorem of Newman says that S3,0(n)> 0 for all n. Inthispaper it is proved that the same assertion holds if q is divisible by 3 or q =4 N +1. On the other hand, it is shown that the number of primes q ≤ x with this property is o(x / log x). Finally, analoga for “higher parities ” are provided. 1.
Minimal Forbidden Words and Symbolic Dynamics
, 1996
"... . We introduce a new complexity measure of a factorial formal language L: the growth rate of the set of minimal forbidden words. We prove some combinatorial properties of minimal forbidden words. As main result we prove that the growth rate of the set of minimal forbidden words for L is a topolo ..."
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. We introduce a new complexity measure of a factorial formal language L: the growth rate of the set of minimal forbidden words. We prove some combinatorial properties of minimal forbidden words. As main result we prove that the growth rate of the set of minimal forbidden words for L is a topological invariant of the dynamical system defined by L. Classification: Automata and Formal Languages 1 Introduction Let L ae A be a factorial language, i.e. a language containing all factors of its words. A word w 2 A is a minimal forbidden word for L if w = 2 L and all proper factors of w belong to L. We denote by MF (L) the language of minimal forbidden words for L. It turns out (as also stressed by the results of this paper) that the combinatorial properties of MF (L) provide an usefull tool to investigate the structure of the language L or of the system that it describes. Consider, for instance, the case of locally testable factorial languages (cf [14]): they are characterized by th...
Pseudopower Avoidance
, 2012
"... Repetition avoidance has been intensely studied since Thue’s work in the early 1900’s. In this paper, we consider another type of repetition, called pseudopower, inspired by the WatsonCrick complementarity property of DNA sequences. A DNA single strand can be viewed as a string over the fourletter ..."
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Repetition avoidance has been intensely studied since Thue’s work in the early 1900’s. In this paper, we consider another type of repetition, called pseudopower, inspired by the WatsonCrick complementarity property of DNA sequences. A DNA single strand can be viewed as a string over the fourletter alphabet{A,C,G,T}, wherein A is the complement of T, while C is the complement of G. Such a DNA single strand will bind to a reverse complement DNA single strand, called its WatsonCrick complement, to form a helical doublestranded DNA molecule. The WatsonCrick complement of a DNA strand is deducible from, and thus informationally equivalent to, the original strand. We use this fact to generalize the notion of the power of a word by relaxing the meaning of “sameness ” to include the image through an antimorphic involution, the model of DNA WatsonCrick complementarity. Given a finite alphabet Σ, an antimorphic involution is a function θ: Σ ∗ −→ Σ ∗ which is an involution, i.e.,θ 2 equals the identity, and an antimorphism, i.e., θ(uv) = θ(v)θ(u), for all u ∈ Σ ∗. For a positive integer k, we call a word w a pseudokthpower with respect to θ if it can be written as w = u1...uk, where for 1 ≤ i,j ≤ k we have either ui = uj or ui = θ(uj). The classical kthpower of a word is a special case of a pseudokthpower, where all the repeating units are identical. We first classify the alphabets Σ and the antimorphic involutions θ for which
On BetaShifts Having Arithmetical Languages
"... Abstract. Let β be a real number with 1 < β < 2. We prove that the language of the βshift is ∆ 0 n iff β is a ∆nreal. The special case where n is 1 is the independently interesting result that the language of the βshift is decidable iff β is a computable real. The “if ” part of the proof is nonc ..."
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Abstract. Let β be a real number with 1 < β < 2. We prove that the language of the βshift is ∆ 0 n iff β is a ∆nreal. The special case where n is 1 is the independently interesting result that the language of the βshift is decidable iff β is a computable real. The “if ” part of the proof is nonconstructive; we show that for Walters ’ version of the βshift, no constructive proof exists. 1
Counting ordered patterns in words generated by morphisms
, 2007
"... We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and that with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, ..."
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We start a general study of counting the number of occurrences of ordered patterns in words generated by morphisms. We consider certain patterns with gaps (classical patterns) and that with no gaps (consecutive patterns). Occurrences of the patterns are known, in the literature, as rises, descents, (non)inversions, squares and prepetitions. We give recurrence formulas in the general case, then deducing exact formulas for particular families of morphisms. Many (classical or new) examples are given illustrating the techniques and showing their interest.
Associative Algebras Satisfying a Semigroup Identity
, 1997
"... Denote by (R; \Delta) the multiplicative semigroup of an associative algebra R over an infinite field, and let (R; ffi) represent R when viewed as a semigroup via the circle operation x ffi y = x + y + xy. In this paper we characterize the existence of an identity in these semigroups in terms of ..."
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Denote by (R; \Delta) the multiplicative semigroup of an associative algebra R over an infinite field, and let (R; ffi) represent R when viewed as a semigroup via the circle operation x ffi y = x + y + xy. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of R. Namely, we prove that the following conditions on R are equivalent: the semigroup (R; ffi) satisfies an identity; the semigroup (R; \Delta) satisfies a reduced identity; and, the associated Lie algebra of R satisfies the Engel condition. When R is finitely generated these conditions are each equivalent to R being upper Lie nilpotent.