Abstract:We prove that the set of all endpoints of the Julia set of
$f(z)=\exp\!(z)-1$
which escape to infinity under iteration of f is not homeomorphic to the rational Hilbert space
$\mathfrak E$
. As a corollary, we show that the set of all points
$z\in \mathbb C$
whose orbits either escape to
$\infty$
or attract to 0 is path-connected. We extend these results to many other functions in the exponential family.
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