Dowdall, Kapovich and Leininger found analogies of Thurston's fibered
cone and Mcmullen's Laurent polynomial that encodes topological
entropy of the resulting maps. We found analogies of their techniques
in other settings, and apply them to the study of the free factor and
free splittings of free groups, and the geometry of finite graphs and
train track maps. I will talk about my recent work with Hyrungryul
Baik and Dongryul Kim, preliminary results of other projects, as well
as potential applications to combinatorics and numerics.