99 research outputs found
Arithmetic Dynamics
This survey paper is aimed to describe a relatively new branch of symbolic
dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic
expansions of reals and vectors that have a "dynamical" sense. This means
precisely that they (semi-) conjugate a given continuous (or
measure-preserving) dynamical system and a symbolic one. The classes of
dynamical systems and their codings considered in the paper involve: (1)
Beta-expansions, i.e., the radix expansions in non-integer bases; (2)
"Rotational" expansions which arise in the problem of encoding of irrational
rotations of the circle; (3) Toral expansions which naturally appear in
arithmetic symbolic codings of algebraic toral automorphisms (mostly
hyperbolic).
We study ergodic-theoretic and probabilistic properties of these expansions
and their applications. Besides, in some cases we create "redundant"
representations (those whose space of "digits" is a priori larger than
necessary) and study their combinatorics.Comment: 45 pages in Latex + 3 figures in ep
On the topology of sums in powers of an algebraic number
Let and It is well known that if is not a root of a polynomial with
coefficients , then is dense in . We give
several sufficient conditions for the denseness of when is a
root of such a polynomial. In particular, we prove that if is not a Perron
number or it has a conjugate such that , then
is dense in .Comment: 10 pages, no figure
Open maps: small and large holes with unusual properties
Let be a two-sided subshift on a finite alphabet endowed with a mixing
probability measure which is positive on all cylinders in . We show that
there exist arbitrarily small finite overlapping union of shifted cylinders
which intersect every orbit under the shift map.
We also show that for any proper subshift of there exists a finite
overlapping unions of shifted cylinders such that its survivor set contains
(in particular, it can have entropy arbitrarily close to the entropy of ).
Both results may be seen as somewhat counter-intuitive.
Finally, we apply these results to a certain class of hyperbolic algebraic
automorphisms of a torus.Comment: 15 pages, no figure
Two-dimensional self-affine sets with interior points, and the set of uniqueness
Let be a real matrix with both eigenvalues less than~1 in
modulus. Consider two self-affine contraction maps from , \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M
v + u, \end{equation*} where . We are interested in the properties of
the attractor of the iterated function system (IFS) generated by
and , i.e., the unique non-empty compact set such that . Our two main results are as follows:
1. If both eigenvalues of are between and in
absolute value, and the IFS is non-degenerate, then has non-empty interior.
2. For almost all non-degenerate IFS, the set of points which have a unique
address is of positive Hausdorff dimension -- with the exceptional cases fully
described as well.
This paper continues our work begun in [11].Comment: 29 pages, 7 figure
A lower bound for the dimension of Bernoulli convolutions
Let and let denote Garsia's entropy for the
Bernoulli convolution associated with . In the present paper
we show that for all and improve this bound
for certain ranges. Combined with recent results by Hochman and
Breuillard-Varj\'u, this yields for all
. In addition, we show that if an algebraic is such that
for some , then
. Such is, for instance, any root of a Pisot number which is
not a Pisot number itself.Comment: 8 pages, no figure
Growth rate for beta-expansions
Let and let m>\be be an integer. Each x\in
I_\be:=[0,\frac{m-1}{\beta-1}] can be represented in the form where
for all (a -expansion of ). It is
known that a.e. has a continuum of distinct -expansions.
In this paper we prove that if is a Pisot number, then for a.e.
this continuum has one and the same growth rate. We also link this rate to the
Lebesgue-generic local dimension for the Bernoulli convolution parametrized by
.
When , we show that the set of -expansions
grows exponentially for every internal .Comment: 21 pages, 2 figure
Multidimensional self-affine sets: non-empty interior and the set of uniqueness
Let be a contracting matrix. In this paper we consider the
self-affine iterated function system , where is a cyclic
vector. Our main result is as follows: if , then the
attractor has non-empty interior.
We also consider the set of points in which have a
unique address. We show that unless belongs to a very special (non-generic)
class, the Hausdorff dimension of is positive. For this special
class the full description of is given as well.
This paper continues our work begun in two previous papers.Comment: 10 pages, no figure
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