Dynamics of composition operators on function spaces defined by local and global properties (Q2673021)

The authors are concerned with composition operators on various locally convex function spaces contained in $C^m(\mathbb R)$. Considered are, for instance, the spaces \[ \mathcal O_n^m(\mathbb R):=\{f\in C^m(\mathbb R): \sup_{x\in\mathbb R} \sup_{0\leq i\leq m}(1+x^2)^{-n}|f^{(i)}(x)|<\infty\}, \] \[ \mathcal O_M(\mathbb R)=\bigcap_{m=1}^\infty \bigcup_{n=1}^\infty \mathcal O_n^m(\mathbb R), \text{ and }\ \mathcal O_C(\mathbb R)=\bigcup_{n=1}^\infty \bigcap_{m=1}^\infty \mathcal O_n^m(\mathbb R), \] under some natural locally convex topologies. Studied are properties such as hypercyclicity, supercyclicity, mixing property, power boundedness, mean ergodicity, and convergence of the iterates in the strong operator topology.