48 research outputs found
Statistic and ergodic properties of Minkowski's diagonal continued fraction
AbstractRecently the author introduced a new class of continued fraction expansions, the S-expansions. Here it is shown that Minkowski's diagonal continued fraction (DCF) is an S-expansion. Due to this, statistic and ergodic properties of the DCF can be given
Entropy quotients and correct digits in number-theoretic expansions
Expansions that furnish increasingly good approximations to real numbers are
usually related to dynamical systems. Although comparing dynamical systems
seems difficult in general, Lochs was able in 1964 to relate the relative speed
of approximation of decimal and regular continued fraction expansions (almost
everywhere) to the quotient of the entropies of their dynamical systems. He
used detailed knowledge of the continued fraction operator. In 2001, a
generalization of Lochs' result was given by Dajani and Fieldsteel in
\citeDajF, describing the rate at which the digits of one number-theoretic
expansion determine those of another. Their proofs are based on covering
arguments and not on the dynamics of specific maps. In this paper we give a
dynamical proof for certain classes of transformations, and we describe
explicitly the distribution of the number of digits determined when comparing
two expansions in integer bases. Finally, using this generalization of Lochs'
result, we estimate the unknown entropy of certain number theoretic expansions
by comparing the speed of convergence with that of an expansion with known
entropy.Comment: Published at http://dx.doi.org/10.1214/074921706000000202 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Metrical theory for -Rosen fractions
The Rosen fractions form an infinite family which generalizes the
nearest-integer continued fractions. In this paper we introduce a new class of
continued fractions related to the Rosen fractions, the -Rosen
fractions. The metrical properties of these -Rosen fractions are
studied. We find planar natural extensions for the associated interval maps,
and show that these regions are closely related to similar region for the
'classical' Rosen fraction. This allows us to unify and generalize results of
diophantine approximation from the literature
Matching of orbits of certain -expansions with a finite set of digits
In this paper we consider a class of continued fraction expansions: the
so-called -expansions with a finite digit set, where is an
integer. These \emph{-expansions with a finite digit set} were introduced in
[KL,L], and further studied in [dJKN,S]. For fixed they are steered by a
parameter . In [KL], for an explicit interval
was determined, such that for all the entropy
of the underlying Gauss-map is equal. In this
paper we show that for all , , such plateaux exist. In
order to show that the entropy is constant on such plateaux, we obtain the
underlying planar natural extension of the maps , the
-invariant measure, ergodicity, and we show that for any two
from the same plateau, the natural extensions are metrically
isomorphic, and the isomorphism is given explicitly. The plateaux are found by
a property called matching
Tong's spectrum for Rosen continued fractions
The Rosen fractions are an infinite set of continued fraction algorithms,
each giving expansions of real numbers in terms of certain algebraic integers.
For each, we give a best possible upper bound for the minimum in appropriate
consecutive blocks of approximation coefficients (in the sense of Diophantine
approximation by continued fraction convergents). We also obtain metrical
results for large blocks of ``bad'' approximations.Comment: 22 pages, 5 figure
(non)-matching and (non)-periodicity for -expansions
Recently a new class of continued fraction algorithms, the
)-expansions, was introduced for each ,
and . Each of these continued fraction algorithms
has only finitely many possible digits. These -expansions `behave'
very different from many other (classical) continued fraction algorithms. In
this paper we will show that when all digits in the digit set are co-prime with
, which occurs in specified intervals of the parameter space, something
extraordinary happens. Rational numbers and certain quadratic irrationals will
not have a periodic expansion. Furthermore, there are no matching intervals in
these regions. This contrasts sharply with the regular continued fraction and
more classical parameterised continued fraction algorithms, for which often
matching is shown to hold for almost every parameter. On the other hand, for
small enough, all rationals have an eventually periodic expansion with
period 1. This happens for all when . We also find infinitely
many matching intervals for , as well as rationals that are not contained
in any matching interval