Thurston's Legacy at the Interface of Surface Dynamics, Train Tracks, and Arithmetic

In this chapter, we discuss the interaction between low-dimensional dynamics, symbolic dynamics, and geometric topology, focusing on the algebraic characterization and asymptotic behaviour of stretch factors or equivalently pseudo-Anosov dilatations. We first review the algebraic foundations linking Perron–Frobenius theory, multi-variable Euclidean algorithms, and Lind’s characterization of topological entropy for shifts of finite type, emphasizing recent constraints on the Perron–Frobenius degree of expansion constants. We then relate these dynamical invariants to the combinatorial world, examining train tracks for surfaces and automorphisms of free groups. Finally, we briefly review the relevance of fibered 3-manifolds, detailing how the Thurston norm, Fried’s concavity theorem over fibered cones, and McMullen’s Teichmüller polynomial come into play.