Clifford Gilmore

Universality of composition operators and applications to complex dynamics

Birkhoff uncovered the first example of a universal operator in 1929, when he showed that translations acting on the space of entire functions can display a maximally divergent behaviour. It was however not until the late 1980s before the investigation of universality received systematic attention. It has subsequently developed into an active area of research that lies at the intersection of topological dynamics, operator theory and ergodic theory.

I will begin with a gentle introduction that recalls the basic definitions and some illustrative examples. I will then consider universality properties of composition operators $C_f \colon g \mapsto g \circ f$, acting on spaces of holomorphic functions, where the symbol $f$ is given by a transcendental entire function restricted to parts of its Fatou set $F(f)$. The investigation of this topic was initiated by Jung (J. d'Analyse Math., 2019) and it lies at the interface of complex dynamics and operator theory.

This is based on joint work with Vasiliki Evdoridou and Myrto Manolaki (JLMS, 2025).