A note on dynamical zeta functions for S-unimodal maps

Dynamical zeta functions for tree maps

We study piecewise monotone and piecewise continuous maps f from a rooted oriented tree to itself, with weight functions either piecewise constant or of bounded variation. We deene kneading coordinates for such tree maps. We show that the Milnor-Thurston relation holds between the weighted reduced zeta function and the weighted kneading determinant of f. This generalizes a result known for piec...

Dynamical Zeta Functions for Maps of the Interval

A dynamical zeta function Ç and a transfer operator Sf are associated with a piecewise monotone map / of the interval [0,1] and a weight function g . The analytic properties of f and the spectral properties of S? are related by a theorem of Baladi and Keller under an assumption of "generating partition". It is shown here how to remove this assumption and, in particular, extend the theorem of Ba...

Equilibrium States for S-unimodal Maps

Dynamical Zeta Functions

The study of periodic orbits for dynamical systems dates back to the very origins of the subject. Given a diieomorphism f : M ! M of a compact manifold it is a classical problem to count the number of periodic orbits fx; fx; : : : ; f n?1 xg, with f n x = x. For each n 1, we denote by N(n) the number of periodic points of period n. General Problem. What can we say about the numbers N(n)? What d...

Dynamical zeta functions

These notes are a rather subjective account of the theory of dynamical zeta functions. They correspond to three lectures presented by the author at the “Numeration” meeting in Leiden in 2010.