Hypercyclicity and universality phenomena with atlas-smooth and atlas-holomorphic sequences

Abstract

Smooth manifolds often require one to account for multiple local coordinate systems. On a smooth manifold like real n-dimensional space, we typically work within a single global coordinate system. Consequently, it is not hard to define partial differentiation operators, for example, and show that they are hypercyclic. However, defining a partial differentiation operator on smooth functions defined globally on general smooth manifolds is difficult due to the multiple local coordinate systems. We introduce the concepts of atlas-smooth and atlas-holomorphic sequences, which we use to study hypercyclicity and universality on spaces of functions defined both locally and globally on manifolds. We focus on partial differentiation operators acting on smooth functions defined on smooth manifolds, and we also consider complex manifolds as well. In 1941, Seidel and Walsh [5] showed that a certain sequence is universal on the space of holomorphic functions defined on the open unit disk. We use the ideas developed here to extend this result to certain complex manifolds.