Escaping points and semiconjugation of holomorphic functions [ Back ]

Date:
29.03.16   
Times:
11:15 to 12:15
Place:
IMUB-Universitat de Barcelona
Speaker:
David Martí
University:
Open University

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 \[ F(z + 2\pi i) = F(z) + ind(f) * 2\pi i \] 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 $\mathbb C^*$.