Abstract
For
$\lambda \in (0,\,1/2]$
let
$K_\lambda \subset \mathbb {R}$
be a self-similar set generated by the iterated function system
$\{\lambda x,\, \lambda x+1-\lambda \}$
. Given
$x\in (0,\,1/2)$
, let
$\Lambda (x)$
be the set of
$\lambda \in (0,\,1/2]$
such that
$x\in K_\lambda$
. In this paper we show that
$\Lambda (x)$
is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any
$y_1,\,\ldots,\, y_p\in (0,\,1/2)$
there exists a full Hausdorff dimensional set of
$\lambda \in (0,\,1/2]$
such that
$y_1,\,\ldots,\, y_p \in K_\lambda$
.
Publisher
Cambridge University Press (CUP)
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