Abstract
We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential$\exp (z)+a$when$a\in (-\infty ,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function$f$, the escaping Julia set$I(f)\cap J(f)$is first category.
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