scholarly journals A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers

2018 ◽  
Vol 371 (5) ◽  
pp. 3281-3308 ◽  
Author(s):  
Yann Bugeaud ◽  
Dong Han Kim
Keyword(s):  
Complexity Function ◽  
Sturmian Words

Sturmian words and overexponential codimension growth

2018 ◽  
Vol 95 ◽  
pp. 53-64
Author(s):  
Antonio Giambruno ◽  
Mikhail Zaicev
Keyword(s):  
Sturmian Words

scholarly journals Sturmian words, Lyndon words and trees

1997 ◽  
Vol 178 (1-2) ◽  
pp. 171-203 ◽  
Author(s):  
Jean Berstel ◽  
Aldo de Luca
Keyword(s):  
Sturmian Words ◽  
Lyndon Words

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

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 A geometrical approach of palindromic factors of standard billiard words

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Jean-Pierre Borel
Keyword(s):  
Geometrical Approach ◽  
International Audience ◽  
Sturmian Words

International audience Many results are already known, concerning the palindromic factors and the palindomic prefixes of Standard billiard words, i.e., Sturmian words and billiard words in any dimension, starting at the origin. We give new geometrical proofs of these results, especially for the existence in any dimension of Standard billiard words with arbitrary long palindromic prefixes.

scholarly journals Some Characterizations of Sturmian Words in Terms of the Lexicographic Order

2012 ◽  
Vol 116 (1-4) ◽  
pp. 25-33 ◽  
Author(s):  
Michelangelo Bucci ◽  
Alessandro De Luca ◽  
Luca Q. Zamboni
Keyword(s):  
Lexicographic Order ◽  
Sturmian Words

On the Index of Sturmian Words

1999 ◽  
pp. 287-294 ◽  
Author(s):  
Jean Berstel
Keyword(s):  
Sturmian Words

scholarly journals Subshifts on Infinite Alphabets and Their Entropy

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1293
Author(s):  
Sharwin Rezagholi
Keyword(s):  
Growth Rate ◽  
Markov Chains ◽  
Topological Entropy ◽  
Spectral Radius ◽  
Symbolic Dynamics ◽  
Infinite Graphs ◽  
Complexity Function ◽  
Countably Infinite

We analyze symbolic dynamics to infinite alphabets by endowing the alphabet with the cofinite topology. The topological entropy is shown to be equal to the supremum of the growth rate of the complexity function with respect to finite subalphabets. For the case of topological Markov chains induced by countably infinite graphs, our approach yields the same entropy as the approach of Gurevich We give formulae for the entropy of countable topological Markov chains in terms of the spectral radius in l2.

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