Finitely many hypercyclic (respectively, supercyclic) operators acting on a
common topological vector space are called disjoint if their direct sum has a hypercyclic (respectively, supercyclic) vector on the diagonal. In this dissertation, we characterize disjointness among hypercyclic and supercyclic linear fractional composition operators on the Hardy space, complementing a celebrated characterization of the cyclic behavior of such operators due to Bourdon and Shapiro [P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997)].
We use our characterization to answer a question by Bernal [L. Bernal-Gonzalez, Disjoint hypercyclic operators, Studia Math. 182 Vol 2 (2007) 113-131, Problem 3],
whether finitely many hypercyclic composition operators on H(D) generated by non-elliptic automorphisms are disjoint.We also apply our characterization to provide N > 1 invertible hypercyclic operators that are disjoint and so that their inverses are not disjoint supercyclic, solving a problem by Bes and Peris [J. Bes and A. Peris, Disjointness in hypercyclicity, J. Math. Anal. Appl. 336 (2007) 297-315, Problem 3].
We also provide characterizations for disjointness of finitely many hypercyclic (respectively, supercyclic) sequences of composition operators with automorphic symbols of any simply connected domain. We show that finitely many sequences of composition operators induced by automorphic symbols are disjoint hypercyclic if and only if they are disjoint supercyclic, complementing and improving recent work by Bernal, Bonilla, and Calderon [L. Bernal-Gonzalez, A. Bonilla and M. C. Calderon-Moreno, Compositional hypercyclicity equals supercyclicity, Houston Journal of Mathematics 3 No 2 (2007) 581-591].
Finally, we characterize disjointness among powers of supercyclic shift operators on ℓp spaces (1 ≤ p, p finite) complementing the study of the hypercyclic case by Bes and Peris.