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The Maximal Subword Complexity of Quasiperiodic Infinite Words

Ronny Polley, Ludwig Staiger
2010 Electronic Proceedings in Theoretical Computer Science  
We provide an exact estimate on the maximal subword complexity for quasiperiodic infinite words.  ...  Our estimate of the subword complexity now follows from this result, previously known results on the subword complexity and elementary results on formal power series.  ...  smallest Pisot-Vijayaraghavan number, that is, the unique real root t P of the cubic polynomial x 3 − x − 1, which is approximately equal to t P ≈ 1.324718.  ... 
doi:10.4204/eptcs.31.19 fatcat:al2hbwy6xbcbvoashckgkbeh3y

The Maximal Complexity of Quasiperiodic Infinite Words

Ludwig Staiger
2021 Axioms  
This allows us to calculate the maximal subword (or factor) complexity of quasiperiodic infinite strings having quasiperiod q and further to derive that maximally complex quasiperiodic infinite strings  ...  It is shown that, for every length l≥3, a word of the form anban (or anbban if l is even) generates the most complex infinite string having this word as quasiperiod.  ...  Conflicts of Interest: The author declares no conflict of interest.  ... 
doi:10.3390/axioms10040306 fatcat:4ywsoa6iajhy7eq4tfyhrimzna

Mechanical Proofs of Properties of the Tribonacci Word [chapter]

Hamoon Mousavi, Jeffrey Shallit
2015 Lecture Notes in Computer Science  
We use it to reprove some old results about the Tribonacci word from the literature, such as assertions about the occurrences in T of squares, cubes, palindromes, and so forth.  ...  This class includes, for example, the famous Tribonacci word T = 0102010010201 · · · , the fixed point of the morphism 0 → 01, 1 → 02, 2 → 0.  ...  The subword complexity function of T is 2n + 1. We now turn to computing the exact number of square occurrences in the finite Tribonacci words Y n .  ... 
doi:10.1007/978-3-319-23660-5_15 fatcat:7schsddrcjbzpedrpjsmd72hqe

Mechanical Proofs of Properties of the Tribonacci Word [article]

Hamoon Mousavi, Jeffrey Shallit
2014 arXiv   pre-print
We use it to reprove some old results about the Tribonacci word from the literature, such as assertions about the occurrences in T of squares, cubes, palindromes, and so forth.  ...  This class includes, for example, the famous Tribonacci word T = 0102010010202 ..., the fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 0.  ...  The subword complexity function of T is 2n + 1. We now turn to computing the exact number of square occurrences in the finite Tribonacci words Y n .  ... 
arXiv:1407.5841v2 fatcat:yerinwdqz5hq7cgrllrbd5quaq

Decision algorithms for Fibonacci-automatic Words, I: Basic results

Hamoon Mousavi, Luke Schaeffer, Jeffrey Shallit
2016 RAIRO - Theoretical Informatics and Applications  
and comparisons, will decide the truth of that proposition.  ...  We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth.  ...  See, for example, the recent papers [2, 31, 33, 32, 34] . Furthermore, in many cases we can explicitly enumerate various aspects of such sequences, such as subword complexity [13] .  ... 
doi:10.1051/ita/2016010 fatcat:bnyljkmxbzcjlnvhz4a5kdbgkq

Decision Algorithms for Fibonacci-Automatic Words, with Applications to Pattern Avoidance [article]

Chen Fei Du, Hamoon Mousavi, Luke Schaeffer, Jeffrey Shallit
2014 arXiv   pre-print
We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth.  ...  This class includes, for example, the famous Fibonacci word f = 01001010..., the fixed point of the morphism 0 -> 01 and 1 -> 0.  ...  Eric Rowland thought about the proof of Theorem 54 with us in 2010, and was able to prove at that time that the word 1213121512131218 · · · avoids additive squares.  ... 
arXiv:1406.0670v4 fatcat:zw7r4zxpkfgtpbyrxnrflxbfvy

Pell and Clapeyron Words as Stable Trajectories in Dynamical Systems [article]

Felix Flicker
2017 arXiv   pre-print
These grow, via a generalization of the period-doubling cascade, as a sequence of stable orbits with periods increasing as the Pell and Clapeyron numbers, providing systematic approximations which can  ...  This Paper proves and extends the results of a companion Letter, as well as providing a pedagogical background.  ...  The requirement for a 2 × 2 substitution matrix to define a quasicrystal is that its eigenvalues are Pisot-Vijayanarayanan numbers: one must be greater than one, and the absolute magnitude of the other  ... 
arXiv:1707.09333v2 fatcat:ebvgcekia5hr7e6j3emuktetae

Lyapunov Exponents in the Spectral Theory of Primitive Inflation Systems

Chrizaldy Neil C. Manibo
2019
Furthermore, we present the recovery of known singularity results and point out connections to number-theoretic quantities which naturally arise from these objects, such as logarithmic Mahler measures.  ...  Using tools from the theory of Lyapunov exponents, we provide a sufficient criterion to rule out the presence of absolutely continuous components in the diffraction and a necessary condition to have a  ...  A substitution is Pisot whenever λ PF of M is a PV number, and is non-Pisot otherwise. An important class of non-Pisot numbers is the set of Salem numbers.  ... 
doi:10.4119/unibi/2935972 fatcat:nfzju7dagzemdn3pkrra3frcji