Connectedness properties of the set where the iterates of an entire function are unbounded. (23rd March 2016)

• Record Type: Journal Article
• Authors: OSBORNE, J. W.; RIPPON, P. J.; STALLARD, G. M.
• Is Part Of: Ergodic theory and dynamical systems. Volume 37:Number 4(2017)
• Page Start: 1291
• Page End: 1307
• Publication Date: 2016-03-23
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2015.85
• Languages: English
• ISSNs: 0143-3857

Abstract

Abstract : We investigate the connectedness properties of the set $I^{\!+\!}(f)$ of points where the iterates of an entire function $f$ are unbounded. In particular, we show that $I^{\!+\!}(f)$ is connected whenever iterates of the minimum modulus of $f$ tend to $\infty$ . For a general transcendental entire function $f$, we show that $I^{\!+\!}(f)\cup \{\infty \}$ is always connected and that, if $I^{\!+\!}(f)$ is disconnected, then it has uncountably many components, infinitely many of which are unbounded.