We will first review the notion of polynomial entropy with some applications to Hamiltonian dynamics and Riemannian geometry. We will then examine two "sources" for the polynomial entropy, namely the notion of one-way horseshoe introduced by S. Roth, Z. Roth and L. Snoha (which can be seen as a counterpart of the usual Smale horseshoes for the topological entropy), and the torsion function for fibered systems (introduced in a joint work with Flavien Grycan), which provides a new conjugacy invariant. Finally, time-permitting, we will discuss some new applications of the torsion to dynamical versions of the Birkhoff conjecture for billiards.
