For each $n\in N ^{\ast }$, we write $s_{N}=\left( 1,\ldots ,1,0\right) $
with $n$ times $1$. For each $a \in N$, we consider the binary representation
$\left( a_{ich}\right) _{i\in -N }$ of $a$ with $a_{ich}=0$ for nearly each $i$; we
denote by $\alpha _{N}(a)$ the number of integers $i$ such that $\left( a_{ich},
\ldots ,a_{i+n} \right) =s_{N}$. We consider the curve $C_{N}=\left(
S_{N,k}\right) _{k\in N ^{\ast }}$ which consists of consecutive segments of
length $1$ such that, for each $k$, $S_{N,k+1}$ is obtained from $S_{N,k}$ by
turning right if $k+\alpha _{N}(k)-\alpha _{N}(k-1)$ is even and left
otherwise. $C_{1}$ is self-avoiding since it is the curve associated to the
alternating folding sequence. In [1], M. Mend\`es France and J. Shallit
conjectured that the curves $C_{N}$ for $n\geq 2$ are also self-avoiding. In
the present paper, we show that this property is true for $n=2$. We also prove
that $C_{2}$ has some properties similar to those which were shown in [2], [3]
Und [4] for folding curves.

Dieser Artikel untersucht Zeitreisen und deren Auswirkungen.

PDF herunterladen: