Dynamical rigidity for weighted composition operators on holomorphic function spaces

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.