Polynomial Entropy and Polynomial Torsion for Fibered Systems

Abstract

Given a continuous fibered dynamical system, we first introduce the notion of polynomial torsion of a fiber, which measures the “infinitesimal variation” of the dynamics between the fiber and the neighboring ones. This gives rise to an (upper semicontinous) torsion function, defined on the base of the system, which is a new \(C^{0}\) (fiber) conjugacy invariant. We prove that the polynomial entropy of the system is the supremum of the torsion of its fibers, which yields a new insight into the creation of polynomial entropy in fibered systems. We examine the relevance of these results in the context of integrable Hamiltonian systems or diffeomorphisms, with the particular cases of \(C^{0}\)-integrable twist maps on the annulus and geodesic flows. Finally, we bound from below the polynomial entropy of \(\ell\)-modal interval maps in terms of their lap number and answer a question by Gomes and Carneiro.