The 3x + 1 Conjugacy Map over a Sturmian Word

Integers ◽  
2009 ◽  
Vol 9 (2) ◽  
Author(s):  
Josefina López ◽  
Peter Stoll
Keyword(s):  
Sturmian Word

AbstractThe 3

scholarly journals The Maximal Number of Runs in Standard Sturmian Words

10.37236/2473 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Paweł Baturo ◽  
Marcin Piątkowski ◽  
Wojciech Rytter
Keyword(s):  
Explicit Formula ◽  
Special Class ◽  
Linear Time ◽  
Infinite Sequence ◽  
Integer Sequence ◽  
Sturmian Word ◽  
Compact Representations ◽  
Sturmian Words ◽  
Standard Word

We investigate some repetition problems for a very special class $\mathcal{S}$ of strings called the standard Sturmian words, which  have very compact representations in terms of sequences of integers. Usually the size of this word is exponential with respect to the size of its integer sequence, hence we are dealing with repetition problems in compressed strings. An explicit formula is given for the number $\rho(w)$ of runs in a standard word $w$. We show that $\rho(w)/|w|\le 4/5$ for each $w\in S$, and  there is an infinite sequence of strictly growing words $w_k\in {\mathcal{S}}$ such that $\lim_{k\rightarrow \infty} \frac{\rho(w_k)}{|w_k|} = \frac{4}{5}$. Moreover, we show how to compute the number of runs in a standard Sturmian word in linear time with respect to the size of its compressed representation.

scholarly journals An Inequality for the Number of Periods in a Word

2021 ◽  
pp. 1-18
Author(s):  
Daniel Gabric ◽  
Narad Rampersad ◽  
Jeffrey Shallit
Keyword(s):  
Critical Exponent ◽  
Infinite Words ◽  
Sturmian Word ◽  
Special Cases

We prove an inequality for the number of periods in a word [Formula: see text] in terms of the length of [Formula: see text] and its initial critical exponent. Next, we characterize all periods of the length-[Formula: see text] prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of [Formula: see text], and use this result to show that our inequality is tight for infinitely many words [Formula: see text]. We propose two related measures of periodicity for infinite words. Finally, we also consider special cases where [Formula: see text] is overlap-free or squarefree.

scholarly journals Sumsets of Finite Beatty Sequences

10.37236/1614 ◽  
2000 ◽  
Vol 8 (2) ◽  
Author(s):  
Jane Pitman
Keyword(s):  
Arithmetic Progression ◽  
Continued Fractions ◽  
Sturmian Word ◽  
Beatty Sequences ◽  
Sturmian Words ◽  
Beatty Sequence

An investigation of the size of $S+S$ for a finite Beatty sequence $S=(s_i)=(\lfloor i\alpha+\gamma \rfloor)$, where $\lfloor \hphantom{x} \rfloor$ denotes "floor", $\alpha$, $\gamma$ are real with $\alpha\ge 1$, and $0\le i \le k-1$ and $k\ge 3$. For $\alpha>2$, it is shown that $|S+S|$ depends on the number of "centres" of the Sturmian word $\Delta S=(s_i-s_{i-1})$, and hence that $3(k-1)\le |S+S|\le 4k-6$ if $S$ is not an arithmetic progression. A formula is obtained for the number of centres of certain finite periodic Sturmian words, and this leads to further information about $|S+S|$ in terms of finite nearest integer continued fractions.

scholarly journals A Square Root Map on Sturmian Words

10.37236/6074 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Jarkko Peltomäki ◽  
Markus A. Whiteland
Keyword(s):  
Fixed Points ◽  
General Setting ◽  
Square Root ◽  
Sturmian Word ◽  
Word Equation ◽  
Sturmian Words ◽  
Infinite Set ◽  
Curious Property

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope $\alpha$, there exists exactly six minimal squares in its language (a minimal square does not have a square as a proper prefix). A Sturmian word $s$ of slope $\alpha$ can be written as a product of these six minimal squares: $s = X_1^2 X_2^2 X_3^2 \cdots$. The square root of $s$ is defined to be the word $\sqrt{s} = X_1 X_2 X_3 \cdots$. The main result of this paper is that $\sqrt{s}$ is also a Sturmian word of slope $\alpha$. Further, we characterize the Sturmian fixed points of the square root map, and we describe how to find the intercept of $\sqrt{s}$ and an occurrence of any prefix of $\sqrt{s}$ in $s$. Related to the square root map, we characterize the solutions of the word equation $X_1^2 X_2^2 \cdots X_n^2 = (X_1 X_2 \cdots X_n)^2$ in the language of Sturmian words of slope $\alpha$ where the words $X_i^2$ are minimal squares of slope $\alpha$.We also study the square root map in a more general setting. We explicitly construct an infinite set of non-Sturmian fixed points of the square root map. We show that the subshifts $\Omega$ generated by these words have a curious property: for all $w \in \Omega$ either $\sqrt{w} \in \Omega$ or $\sqrt{w}$ is periodic. In particular, the square root map can map an aperiodic word to a periodic word.

scholarly journals The sequence of open and closed prefixes of a Sturmian word

2017 ◽  
Vol 90 ◽  
pp. 27-45 ◽  
Author(s):  
Alessandro De Luca ◽  
Gabriele Fici ◽  
Luca Q. Zamboni
Keyword(s):  
Sturmian Word

scholarly journals Modified complexity and $*$-Sturmian word

1999 ◽  
Vol 75 (3) ◽  
pp. 26-28 ◽  
Author(s):  
Izumi Nakashima ◽  
Jun-ichi Tamura ◽  
Shin-ichi Yasutomi
Keyword(s):  
Sturmian Word

scholarly journals Counting Lyndon Factors

10.37236/6915 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Amy Glen ◽  
Jamie Simpson ◽  
W. F. Smyth
Keyword(s):  
Lower Bound ◽  
Golden Ratio ◽  
Fibonacci Number ◽  
Expected Number ◽  
Open Problems ◽  
Sturmian Word ◽  
Minimum Number ◽  
Lyndon Word ◽  
Lyndon Words

In this paper, we determine the maximum number of distinct Lyndon factors that a word of length $n$ can contain. We also derive formulas for the expected total number of Lyndon factors in a word of length $n$ on an alphabet of size $\sigma$, as well as the expected number of distinct Lyndon factors in such a word. The minimum number of distinct Lyndon factors in a word of length $n$ is $1$ and the minimum total number is $n$, with both bounds being achieved by $x^n$ where $x$ is a letter. A more interesting question to ask is what is the minimum number of distinct Lyndon factors in a Lyndon word of length $n$? In this direction, it is known (Saari, 2014) that a lower bound for the number of distinct Lyndon factors in a Lyndon word of length $n$ is $\lceil\log_{\phi}(n) + 1\rceil$, where $\phi$ denotes the golden ratio $(1 + \sqrt{5})/2$. Moreover, this lower bound is sharp when $n$ is a Fibonacci number and is attained by the so-called finite Fibonacci Lyndon words, which are precisely the Lyndon factors of the well-known infinite Fibonacci word $\boldsymbol{f}$ (a special example of an infinite Sturmian word). Saari (2014) conjectured that if $w$ is Lyndon word of length $n$, $n\ne 6$, containing the least number of distinct Lyndon factors over all Lyndon words of the same length, then $w$ is a Christoffel word (i.e., a Lyndon factor of an infinite Sturmian word). We give a counterexample to this conjecture. Furthermore, we generalise Saari's result on the number of distinct Lyndon factors of a Fibonacci Lyndon word by determining the number of distinct Lyndon factors of a given Christoffel word. We end with two open problems.

scholarly journals Words with many Palindrome Pair Factors

10.37236/5583 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Adam Borchert ◽  
Narad Rampersad
Keyword(s):  
Infinite Words ◽  
Sturmian Word ◽  
Proper Factor ◽  
Sturmian Words

Motivated by a conjecture of Frid, Puzynina, and Zamboni, we investigate infinite words with the property that for infinitely many $n$, every length-$n$ factor is a product of two palindromes. We show that every Sturmian word has this property, but this does not characterize the class of Sturmian words. We also show that the Thue—Morse word does not have this property. We investigate finite words with the maximal number of distinct palindrome pair factors and characterize the binary words that are not palindrome pairs but have the property that every proper factor is a palindrome pair.

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