158 research outputs found
On the Number of Unbordered Factors
We illustrate a general technique for enumerating factors of k-automatic
sequences by proving a conjecture on the number f(n) of unbordered factors of
the Thue-Morse sequence. We show that f(n) = 4 and that f(n) = n
infinitely often. We also give examples of automatic sequences having exactly 2
unbordered factors of every length
Resolvent Methods for Quantum Walks with an Application to a Thue-Morse Quantum Walk
In this expository note, we discuss spatially inhomogeneous quantum walks in
one dimension and describe a genre of mathematical methods that enables one to
translate information about the time-independent eigenvalue equation for the
unitary generator into dynamical estimates for the corresponding quantum walk.
To illustrate the general methods, we show how to apply them to a 1D coined
quantum walk whose coins are distributed according to an element of the
Thue--Morse subshift.Comment: This paper is part of the proceedings volume for the Workshop on
"Quantum Simulation and Quantum Walks" held in Yokohama, Japan in November of
201
Automatic sequences as good weights for ergodic theorems
We study correlation estimates of automatic sequences (that is, sequences
computable by finite automata) with polynomial phases. As a consequence, we
provide a new class of good weights for classical and polynomial ergodic
theorems, not coming themselves from dynamical systems.
We show that automatic sequences are good weights in for polynomial
averages and totally ergodic systems. For totally balanced automatic sequences
(i.e., sequences converging to zero in mean along arithmetic progressions) the
pointwise weighted ergodic theorem in holds. Moreover, invertible
automatic sequences are good weights for the pointwise polynomial ergodic
theorem in , .Comment: 31 page
Subword complexity and power avoidance
We begin a systematic study of the relations between subword complexity of
infinite words and their power avoidance. Among other things, we show that
-- the Thue-Morse word has the minimum possible subword complexity over all
overlap-free binary words and all -power-free binary words, but not
over all -power-free binary words;
-- the twisted Thue-Morse word has the maximum possible subword complexity
over all overlap-free binary words, but no word has the maximum subword
complexity over all -power-free binary words;
-- if some word attains the minimum possible subword complexity over all
square-free ternary words, then one such word is the ternary Thue word;
-- the recently constructed 1-2-bonacci word has the minimum possible subword
complexity over all \textit{symmetric} square-free ternary words.Comment: 29 pages. Submitted to TC
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced
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