Johan Nilsson
The Fine Structure of Dyadically Badly Approximable Numbers.
Thu, 02/25/2010 - 16:01 — SomNambulAuthors: Johan Nilsson
Subjects: Dynamical Systems
AbstractWe consider badly approximable numbers in the case of dyadic diophantine
approximation. For the unit circle $\mathbb{S}$ and the smallest distance to an
integer $\|\cdot\|$ we give elementary proofs that the set $F(c) = \{x \in
\mathbb{S}: \|2^nx\| \geq c, n\geq 0\}$ is a fractal set whose Hausdorff
dimension depends continuously on $c$, is constant on intervals which form a
set of Lebesgue measure 1 and is self-similar. Hence it has a fractal graph.
Moreover, the dimension of $F(c)$ is zero if and only if $c\geq 1-2\tau$, where
$\tau$ is the Thue-Morse constant.On the Entropy of Random Fibonacci Words.
Thu, 01/21/2010 - 16:01 — SomNambulAuthors: Johan Nilsson
Subjects: Combinatorics
AbstractThe random Fibonacci chain is a generalisation of the classical Fibonacci
substitution and is defined as the rule mapping $0\mapsto 1$ and $1 \mapsto 01$
with probability $p$ and $1 \mapsto 10$ with probability $1-p$ for $0<p<1$ and
where the random rule is applied each time it acts on a 1. We show that the
topological entropy of this object is given by the growth rate of the set of
inflated random Fibonacci words.
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