In causal set theory, cycles of cosmic expansion and collapse are modelled by causal sets with 'breaks' and 'posts' and a special role is played by cyclic dynamics in which the universe goes through perpetual cycles. We identify and characterise two algebras of observables for cyclic dynamics in which the causal set universe has infinitely many breaks. The first algebra is constructed from the cylinder sets associated with finite causal sets that have a single maximal element and offers a new framework for defining cyclic dynamics as random walks on a novel tree. The second algebra is generated by a collection of stem-sets and offers a physical interpretation of the observables in these models as statements about unlabelled stems with a single maximal element. There are analogous theorems for cyclic dynamics in which the causal set universe has infinitely many posts.