Commentary on “Differentiable dynamical systems” by Stephen Smale

Abstract

Abstract. Immediately following the commentary below, this previously published article is reprinted in its entirety: Stephen Smale, "Differentiable dynamical systems", Bull. Amer. Math. Soc. 73 (1967), no. 6, 747-817. It would be difficult, and impossible for me, to describe the full impact of Smale's article on differentiable dynamical systems. I will concentrate on one topic from Part II of the paper: dynamical zeta functions for flows. There has been significant improvement in our understanding of these zeta functions in the last few years and many advances culminated in Dyatlov and Guillarmou's [DyGu18] resolution of Smale's conjecture about their meromorphic continuation.To set things up, a flow on a compact smooth manifold M is a one-parameter group of diffeomorphisms:It is generated by a smooth vector field X(x) := dϕ t (x)/dt| t=0 . Let Γ denote the set of closed orbits of the flow, and for γ ∈ Γ let (γ) be the minimal period of γ; that is, the first t > 0 such that ϕ t (x) = x for some x ∈ γ.A dynamical zeta function for flows is defined by Smale in [Sm67, §II.4] as follows:This is a generalization of a zeta function defined by Selberg when the flow is the geodesic flow on a Riemann surface, but in this generality Smale considered a zeta function as "a wild idea". A closely related zeta function was later introduced by Ruelle [Ru76]:.If convergence of the product in the definition of Z(s) is known for Re s 1, then meromorphic continuation of one zeta function follows from that of the other.Needless to say, some assumptions are required to make sense of equation (1), discreteness of the set of (γ) being the first requirement. Convergence for Re s 1 follows from knowing that |{γ : (γ) ≤ T }| ≤ Ce CT . Refining such estimates to obtain prime geodesic theorems is one of the applications of dynamical zeta functions; see [GLP13] and references given therein.

Dynamical torsion for contact Anosov flows

“…Using this microlocal approach, one could try to address questions relating topology and dynamical zeta functions. Quoting the commentary from Zworski [Zwo18] on Smale's seminal paper [Sma67], equation (1.1) "would link dynamical, spectral and topological quantities. [.…”