scholarly journals Extremal Infinite Overlap-Free Binary Words

10.37236/1365 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Jean-Paul Allouche ◽  
James Currie ◽  
Jeffrey Shallit
Keyword(s):  
Fixed Point ◽  
Binary Word ◽  
Infinite Word

Let $\overline{\bf t}$ be the infinite fixed point, starting with $1$, of the morphism $\mu: 0 \rightarrow 01$, $1 \rightarrow 10$. An infinite word over $\lbrace 0, 1 \rbrace$ is said to be overlap-free if it contains no factor of the form $axaxa$, where $a \in \lbrace 0,1 \rbrace$ and $x \in \lbrace 0,1 \rbrace^*$. We prove that the lexicographically least infinite overlap-free binary word beginning with any specified prefix, if it exists, has a suffix which is a suffix of $\overline{\bf t}$. In particular, the lexicographically least infinite overlap-free binary word is $001001 \overline{\bf t}$.

scholarly journals For each $\alpha > 2$ there is an Infinite Binary Word with Critical Exponent $\alpha$

10.37236/909 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
James D. Currie ◽  
Narad Rampersad
Keyword(s):  
Critical Exponent ◽  
Rational Numbers ◽  
Alpha Power ◽  
Binary Word ◽  
Infinite Word ◽  
Open Question

The critical exponent of an infinite word ${\bf w}$ is the supremum of all rational numbers $\alpha$ such that ${\bf w}$ contains an $\alpha$-power. We resolve an open question of Krieger and Shallit by showing that for each $\alpha > 2$ there is an infinite binary word with critical exponent $\alpha$.

On maximal pattern complexity of some automatic words

2010 ◽  
Vol 31 (5) ◽  
pp. 1463-1470 ◽  
Author(s):  
TETURO KAMAE ◽  
PAVEL V. SALIMOV
Keyword(s):  
Fixed Point ◽  
Linear Function ◽  
Finite Subset ◽  
Constant Length ◽  
Sequence Entropy ◽  
Pattern Complexity ◽  
Infinite Words ◽  
Maximum Value ◽  
Infinite Word

AbstractThe pattern complexity of a word for a given pattern S, where S is a finite subset of {0,1,2,…}, is the number of distinct restrictions of the word to S+n (with n=0,1,2,…). The maximal pattern complexity of the word, introduced in the paper of T. Kamae and L. Zamboni [Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys.22(4) (2002), 1191–1199], is the maximum value of the pattern complexity of S with #S=k as a function of k=1,2,…. A substitution of constant length on an alphabet is a mapping from the alphabet to finite words on it of constant length not less than two. An infinite word is called a fixed point of the substitution if it stays the same after the substitution is applied. In this paper, we prove that the maximal pattern complexity of a fixed point of a substitution of constant length on {0,1} (as a function of k=1,2,…) is either bounded, a linear function of k, or 2k.

A NEW COMPLEXITY FUNCTION FOR WORDS BASED ON PERIODICITY

2013 ◽  
Vol 23 (04) ◽  
pp. 963-987 ◽  
Author(s):  
FILIPPO MIGNOSI ◽  
ANTONIO RESTIVO
Keyword(s):  
The Other ◽  
Factorization Theorem ◽  
Complexity Measures ◽  
Infinite Words ◽  
Average Value ◽  
Complexity Function ◽  
Binary Word ◽  
Infinite Word

Motivated by the extension of the critical factorization theorem to infinite words, we study the (local) periodicity function, i.e. the function that, for any position in a word, gives the size of the shortest square centered in that position. We prove that this function characterizes any binary word up to exchange of letters. We then introduce a new complexity function for words (the periodicity complexity) that, for any position in the word, gives the average value of the periodicity function up to that position. The new complexity function is independent from the other commonly used complexity measures as, for instance, the factor complexity. Indeed, whereas any infinite word with bounded factor complexity is periodic, we will show a recurrent non-periodic word with bounded periodicity complexity. Further, we will prove that the periodicity complexity function grows as Θ( log n) in the case of the Fibonacci infinite word and that it grows as Θ(n) in the case of the Thue–Morse word. Finally, we will show examples of infinite recurrent words with arbitrary high periodicity complexity.

scholarly journals Infinite special branches in words associated with beta-expansions

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Christiane Frougny ◽  
Zuzana Masáková ◽  
Edita Pelantová
Keyword(s):  
Fixed Point ◽  
Real Number ◽  
Pisot Number ◽  
Infinite Word ◽  
International Audience ◽  
Pisot Unit

International audience A Parry number is a real number β > 1 such that the Rényi β-expansion of 1 is finite or infinite eventually periodic. If this expansion is finite, β is said to be a simple Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fixed point uβ of the canonical substitution associated with β-expansions, when β is a simple Parry number. In this paper we consider the case where β is a non-simple Parry number. We determine the structure of infinite left special branches, which are an important tool for the computation of the complexity of uβ. These results allow in particular to obtain the following characterization: the infinite word uβ is Sturmian if and only if β is a quadratic Pisot unit.

scholarly journals SUBSTITUTIONS OVER INFINITE ALPHABET GENERATING (−β)-INTEGERS

2012 ◽  
Vol 23 (08) ◽  
pp. 1627-1639
Author(s):  
DANIEL DOMBEK
Keyword(s):  
Fixed Point ◽  
General Setting ◽  
Finite Alphabet ◽  
Infinite Word ◽  
Numeration Systems ◽  
Negative Base

We study positional numeration systems with negative base called (−β)-expansions in a more general setting than that of Ito and Sadahiro. We give an admissibility criterion for (−β)-expansions and discuss the properties of the set of (−β)-integers, denoted by ℤ−β. We give a description of distances between consecutive (−β)-integers and show that ℤ−β can be coded by an infinite word over an infinite alphabet, which is a fixed point of a non-erasing non-trivial morphism. We give a set of examples where ℤ−β is coded by an infinite word over a finite alphabet.

Mental equilibrium: Identifying fixed-point attractors in the stream of thought

2003 ◽  
Author(s):  
Robin R. Vallacher ◽  
Andrzej Nowak ◽  
Matthew Rockloff
Keyword(s):  
Fixed Point

Bifurcation in a Simple Model of the Cardiovascular System

2000 ◽  
Vol 39 (02) ◽  
pp. 118-121 ◽  
Author(s):  
S. Akselrod ◽  
S. Eyal
Keyword(s):  
Nonlinear Dynamics ◽  
Blood Pressure ◽  
Heart Rate ◽  
Fixed Point ◽  
Cardiovascular System ◽  
Modified Model ◽  
Mayer Waves ◽  
Sustained Oscillations ◽  
Human Cardiovascular System ◽  
Physiological Values

Abstract:A simple nonlinear beat-to-beat model of the human cardiovascular system has been studied. The model, introduced by DeBoer et al. was a simplified linearized version. We present a modified model which allows to investigate the nonlinear dynamics of the cardiovascular system. We found that an increase in the -sympathetic gain, via a Hopf bifurcation, leads to sustained oscillations both in heart rate and blood pressure variables at about 0.1 Hz (Mayer waves). Similar oscillations were observed when increasing the -sympathetic gain or decreasing the vagal gain. Further changes of the gains, even beyond reasonable physiological values, did not reveal another bifurcation. The dynamics observed were thus either fixed point or limit cycle. Introducing respiration into the model showed entrainment between the respiration frequency and the Mayer waves.

Sign in / Sign up

Export Citation Format

Share Document