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Fractals in the large |
Canad. J. Math. 50(1998), 638-657
Published:1998-06-01
Printed: Jun 1998
Printed: Jun 1998
- Robert S. Strichartz
Abstract
A {\it reverse iterated function system} (r.i.f.s.) is defined to be a
set of expansive maps
$\{T_1,\ldots,T_m\}$ on a discrete metric space $M$. An invariant set
$F$ is defined to be a set satisfying
$F = \bigcup^m_{j=1} T_jF$, and an invariant measure $\mu$ is
defined to be a solution of
$\mu = \sum^m_{j=1} p_j\mu\circ T_j^{-1}$ for positive weights
$p_j$. The structure and basic properties of such invariant sets
and measures is described, and some examples are given.
A {\it blowup} $\cal F$ of a self-similar set $F$ in
$\Bbb R^n$ is defined to be the union of an increasing sequence of
sets, each similar to $F$. We give a general construction of
blowups, and show that under certain hypotheses a blowup is the sum set of
$F$ with an invariant set for a r.i.f.s. Some examples of blowups of
familiar fractals are described. If $\mu$ is an invariant measure
on $\Bbb Z^+$ for a linear r.i.f.s., we describe the behavior of its
{\it analytic} transform, the power series
$\sum^\infty_{n=0} \mu(n)z^n$ on the unit disc.
| MSC Classifications: | 28A80 - Fractals [See also 37Fxx] |