Rowmotion on the chain of V's poset and whirling dynamics

Given a finite poset P, we study the _whirling_ action on vertex-labelings of P with the elements \0,1,2, ,k\. When such labelings are (weakly) order-reversing, we call them k-bounded P-partitions. We give a general equivariant bijection between k-bounded P-partitions and order ideals of the poset P× [k] which conveys whirling to the well-studied rowmotion operator. As an application, we derive periodicity and homomesy results for rowmotion acting on the chain of V's poset V × [k]. We are able to generalize some of these results to the more complicated dynamics of rowmotion on Cₙ× [k], where Cₙ is the claw poset with n unrelated elements each covering 0.