An important problem in the dynamics of surface homeomorphisms is determining the forcing relation between orbits. The forcing relation between periodic orbits can be computed using standard algorithms, though this does not give much information on the structure of the forcing relation. Here we consider forcing relations between homoclinic orbits, and their relationships with periodic orbits. W...

Every definable forcing class Γ gives rise to a corresponding forcing modality, for which Γ φ means that φ is true in all Γ extensions, and the valid principles of Γ forcing are the modal assertions that are valid for this forcing interpretation. For example, [9] shows that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of t...

An r-fold analogue of the positive semidefinite zero forcing process that is carried out on the r-blowup of a graph is introduced and used to define the fractional positive semidefinite forcing number. Properties of the graph blowup when colored with a fractional positive semidefinite forcing set are examined and used to define a three-color forcing game that directly computes the fractional po...