Bounded composition operators on functional quasi-Banach spaces and stability of dynamical systems

TL;DR

This paper explores the conditions under which composition operators are bounded on certain quasi-Banach spaces of smooth and entire functions, revealing that only affine maps induce bounded operators in these contexts.

Contribution

It establishes that bounded composition operators on these spaces are highly restrictive, allowing only affine maps or specific polynomial automorphisms, thus linking operator boundedness to dynamical system properties.

Findings

Only affine maps induce bounded composition operators on spaces of entire functions in one variable.

Polynomial automorphisms other than affine transforms cannot induce bounded operators in the two-dimensional case.

Boundedness of composition operators imposes strong restrictions on the behavior of the inducing maps.

Abstract

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 a composition operator strongly restricts the behavior of the original map, and it provides an effective method to investigate the properties of composition operators using the theory of dynamical system. Consequently, we prove that only affine maps can induce bounded composition operators on any quasi-Banach space continuously included in the space of entire functions of one variable if the function space contains a nonconstant function. We also prove that any polynomial automorphisms except affine transforms cannot induce bounded composition operators on a quasi-Banach space composed of entire functions in the two-dimensional complex affine space under several mild conditions.