41 research outputs found
On the expansions of real numbers in two integer bases
Let and be multiplicatively independent positive integers. We
establish that the -ary expansion and the -ary expansion of an irrational
real number, viewed as infinite words on and , respectively, cannot have simultaneously a low block
complexity. In particular, they cannot be both Sturmian words.Comment: 11 pages, to appear at Annales de l'Institut Fourie
On the expansions of real numbers in two multiplicative dependent bases
Let and be multiplicatively dependent integers. We
establish a lower bound for the sum of the block complexities of the -ary
expansion and of the -ary expansion of an irrational real number, viewed as
infinite words on and , and we
show that this bound is best possible.Comment: 15pages. arXiv admin note: substantial text overlap with
arXiv:1512.0693
Morphisms preserving the set of words coding three interval exchange
Any amicable pair \phi, \psi{} of Sturmian morphisms enables a construction
of a ternary morphism \eta{} which preserves the set of infinite words coding
3-interval exchange. We determine the number of amicable pairs with the same
incidence matrix in and we study incidence matrices associated
with the corresponding ternary morphisms \eta.Comment: 16 page
On the complexity of algebraic numbers II. Continued fractions
The continued fraction expansion of an irrational number is
eventually periodic if and only if is a quadratic irrationality.
However, very little is known regarding the size of the partial quotients of
algebraic real numbers of degree at least three. Because of some numerical
evidence and a belief that these numbers behave like most numbers in this
respect, it is often conjectured that their partial quotients form an unbounded
sequence. More modestly, we may expect that if the sequence of partial
quotients of an irrational number is, in some sense, "simple", then
is either quadratic or transcendental. The term "simple" can of course
lead to many interpretations. It may denote real numbers whose continued
fraction expansion has some regularity, or can be produced by a simple
algorithm (by a simple Turing machine, for example), or arises from a simple
dynamical system... The aim of this paper is to present in a unified way
several new results on these different approaches of the notion of
simplicity/complexity for the continued fraction expansion of algebraic real
numbers of degree at least three
Matrices of 3iet preserving morphisms
We study matrices of morphisms preserving the family of words coding
3-interval exchange transformations. It is well known that matrices of
morphisms preserving sturmian words (i.e. words coding 2-interval exchange
transformations with the maximal possible factor complexity) form the monoid
, where
.
We prove that in case of exchange of three intervals, the matrices preserving
words coding these transformations and having the maximal possible subword
complexity belong to the monoid $\{\boldsymbol{M}\in\mathbb{N}^{3\times 3} |
\boldsymbol{M}\boldsymbol{E}\boldsymbol{M}^T = \pm\boldsymbol{E},\
\det\boldsymbol{M}=\pm 1\}\boldsymbol{E} =
\Big(\begin{smallmatrix}0&1&1 -1&0&1 -1&-1&0\end{smallmatrix}\Big)$.Comment: 26 pages, 4 figure