18,049 research outputs found
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
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
Dirichlet uniformly well-approximated numbers
Fix an irrational number . For a real number , consider the
numbers satisfying that for all large number , there exists an integer
, such that , where is the
distance of a real number to its nearest integer. These numbers are called
Dirichlet uniformly well-approximated numbers. For any , the Haussdorff
dimension of the set of these numbers is obtained and is shown to depend on the
Diophantine property of . It is also proved that with respect to
, the only possible discontinuous point of the Hausdorff dimension is
.Comment: 35 page
Bounded type interval exchange maps
Irrational numbers of bounded type have several equivalent characterizations.
They have bounded partial quotients in terms of arithmetic characterization and
in the dynamics of the circle rotation, the rescaled recurrence time to
-ball of the initial point is bounded below. In this paper, we consider how
the bounded type condition of irrational is generalized into interval exchange
maps.Comment: 12 page
The dynamical Borel-Cantelli lemma and the waiting time problems
We investigate the connection between the dynamical Borel-Cantelli and
waiting time results. We prove that if a system has the dynamical
Borel-Cantelli property, then the time needed to enter for the first time in a
sequence of small balls scales as the inverse of the measure of the balls.
Conversely if we know the waiting time behavior of a system we can prove that
certain sequences of decreasing balls satisfies the Borel-Cantelli property.
This allows to obtain Borel-Cantelli like results in systems like axiom A and
generic interval exchanges.Comment: In this revision some small errors are correcte
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