Abstract:

For every holomorphic self-map of the punctured plane $f$, there exists an entire funtion $F$ that is semiconjugated to $f$ by the exponential function - we say that $F$ is a lift of $f$. Each holomorphic self-map of $mathbb C^*$ $f$ has an associated index, $ind(f)$, which is an integer such that [ F(z + 2pi i) = F(z) + ind(f) * 2pi i ] for all $z$. We show that if $f$ is a transcendental entire function with no zeros, then the fast escaping set of a lift $F$ of $f$ equals the preimage under the exponential of the fast escaping set of $f$. Bergweiler and Hinkkanen proved one of the inclusions in a more general setting, but we show that equality holds in this particular case. Moreover, we can compare the escaping set, the set of unbounded non-escaping orbits and the set of bounded orbits of $f$ with those of a lift $F$ of $f$ in terms of the index of $f$. Similar results hold for general holomorphic self-maps of $mathbb C^*$.