Dynamical rigidity for weighted composition operators on holomorphic function spaces

TL;DR

This paper investigates how weighted composition operators on holomorphic function spaces are constrained by local dynamics, leading to rigidity results that force the symbol to be affine under various conditions.

Contribution

It introduces a new approach based on local holomorphic dynamics at periodic points, establishing affine-symbol rigidity for broad classes of function spaces.

Findings

Boundedness implies the symbol must be affine in many cases.

Local periodic-point obstructions relate to supercyclicity, hypercyclicity, and cyclicity.

Higher-dimensional rigidity results are established under mild stability assumptions.

Abstract

We study weighted composition operators on quasi-Banach spaces of holomorphic functions via their induced action on jets along periodic orbits. Under a natural graded nondegeneracy condition, boundedness and compactness, together with a nonvanishing condition on the weight along the periodic orbit, impose strong restrictions on the local holomorphic dynamics of the symbol. We also obtain local periodic-point obstructions from supercyclicity, hypercyclicity, and cyclicity. As consequences, we obtain affine-symbol rigidity for bounded weighted composition operators on spaces of entire functions. In one complex variable, if the ambient function space is any infinite-dimensional quasi-Banach space continuously embedded in the space of entire functions, then boundedness forces the symbol to be affine. In particular, this applies to every infinite-dimensional reproducing kernel Hilbert space of entire functions. We also prove a higher-dimensional affine-rigidity theorem under mild stability assumptions, and a weighted rigidity theorem for polynomial automorphisms of two complex variables. Our approach relies on local holomorphic dynamics at periodic points rather than reproducing-kernel formulas or space-specific norm estimates, and it applies uniformly across broad classes of holomorphic function spaces.