On bounded composition operators on functional quasi-Banach spaces and a stability of dynamical systems.

In this paper, we investigate the boundedness of composition operators defined on a quasi-Banach space continuously included in the space of smooth functions on a manifold. We prove that the boundedness of composition operators strongly limits the behavior of the original map defined on the smooth manifold. As a result, we prove that only affine maps can induce a bounded composition operator on any Banach space continuously included in the space of entire functions on the complex plane, in particular, on any reproducing kernel Hilbert space (RKHS) composed of entire functions on the complex plane. We also discuss higher-dimensional cases and prove any polynomial automorphisms except affine transforms cannot induce a bounded composition operator on a Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.