84 research outputs found
Invariant measures, matching and the frequency of 0 for signed binary expansions
We introduce a parametrised family of maps ,
called symmetric doubling maps, defined on by ,
where . Each map generates binary expansions with
digits , 0 and 1. We study the frequency of the digit 0 in typical
expansions as a function of the parameter . The transformations
have a natural ergodic invariant measure that is absolutely
continuous with respect to Lebesgue measure. The frequency of the digit 0 is
related to the measure by the Ergodic Theorem.
We show that the density of is piecewise smooth except for a set of
parameters of zero Lebesgue measure and full Hausdorff dimension and give a
full description of the structure of the maximal parameter intervals on which
the density is piecewise smooth. We give an explicit formula for the frequency
of the digit 0 in typical signed binary expansions on each of these parameter
intervals and show that this frequency depends continuously on the parameter
. Moreover, it takes the value only on the interval and it is strictly less than on the remainder
of the parameter space.Comment: 30 pages, 4 figure
On the points without universal expansions
Let . Given any , a sequence
is called a -expansion of if
For any and any , if there exists some such that
, then we call a
universal -expansion of .
Sidorov \cite{Sidorov2003}, Dajani and de Vries \cite{DajaniDeVrie} proved
that given any , then Lebesgue almost every point has uncountably
many universal expansions. In this paper we consider the set of
points without universal expansions. For any , let be the
-bonacci number satisfying the following equation:
Then we have
, where denotes the Hausdorff dimension.
Similar results are still available for some other algebraic numbers. As a
corollary, we give some results of the Hausdorff dimension of the survivor set
generated by some open dynamical systems. This note is another application of
our paper \cite{KarmaKan}.Comment: 15page
Invariant densities for random -expansions
Let be a non-integer. We consider expansions of the form
, where the digits are
generated by means of a Borel map defined on . We show existence and uniqueness of an
absolutely continuous -invariant probability measure w.r.t. , where is the Bernoulli measure on with
parameter and is the normalized Lebesgue measure on
. Furthermore, this measure is of the
form , where is equivalent with
. We establish the fact that the measure of maximal entropy and are mutually singular. In case the number 1 has a finite
greedy expansion with positive coefficients, the measure is Markov. In the last section we answer a question concerning
the number of universal expansions, a notion introduced in [EK]
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
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
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