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Expansions in non-integer bases: lower, middle and top orders
Let ; it is known that each has an expansion of
the form with . It was shown in
\cite{EJK} that if , then each has a
continuum of such expansions; however, if , then there exist
infinitely many having a unique expansion \cite{GS}.
In the present paper we begin the study of parameters for which there
exists having a fixed finite number of expansions in base . In
particular, we show that if , then each has either 1 or
infinitely many expansions, i.e., there are no such in
.
On the other hand, for each there exists \ga_m>0 such that for any
q\in(2-\ga_m,2), there exists which has exactly expansions in base
.Comment: 15 pages; to appear in J. Number Theor
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
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