Dynamical Zeta Functions and Transfer Operators, Volume 49, Number 8

C ertain generating functions—encoding properties of objects like prime numbers, periodic orbits, ...—have received the name of zeta functions. They are useful in studying the statistical properties of the objects in question. Zeta functions have generally been associated with problems of arithmetic or algebra and tend to have common features: meromorphy, Euler product formula, functional equation, location of poles and zeros (Dirichlet series expansion, Riemann hypothesis), and relation with certain operators (typically operators acting on cohomology groups). The dynamical zeta functions to be discussed here are set up to count periodic orbits but to count them with fairly general weights. As a consequence the subject will have a more function-theoretic flavor than the study of arithmetic or algebraic zeta functions. Apart from that, our zeta functions will have properties similar to those of the more traditional ones. The main difference will be that the relevant operators (called transfer operators) will act on (infinite-dimensional) cochain groups instead of (finite-dimensional) cohomology groups. Intuitively, the weights that we have introduced prevent passage from cochains to cohomology groups. Technically this will force us to consider determinants in infinite dimension. The study of dynamical zeta functions uses original tools (transfer operators, kneading determinants), which we shall discuss below. The simplest invariant measures for a dynamical system are those carried by periodic orbits. Counting periodic orbits is thus a natural task from the point of view of ergodic theory. And dynamical zeta functions are an effective tool to do the counting. The tool turns out to be so effective in fact as to make one suspect that there is more to the story than what we currently understand.