Davis Ruelle: Dynamical Zeta Functions For Maps on the Interval

30-1994; April 1994; Vol 2, pp.212-214

Dynamical Zeta Functions for Maps of the Interval

David Ruelle

Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France
January 18, 1992

Primary 58F20, 58F03; Secondary 58F11

A dynamical zeta function z and a transfer operator L are associated with a piecewise monotone map ¶ of the interval [0,1] and a weight function g. The analytic properties of z and the spectral properties of L 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 Baladi and Keller to the case when ¶ has negative Schwarzian derivative.

Let 0 = a₀ < a₁ < º < a_N = 1. We write X = [0,1] Ã R and assume that ¶ is continuous X Æ X and strictly monotone on the intervals J_i = [ a_i
{1, a_i]. Furthermore, let g : X Æ C have bounded variation. A transfer operator L acting on functions ': X Æ C of bounded variation is defined by

L'(x) =		g(y) '(y),
	à
	y: fy=x

and we let

lim

m Æ 1

sup

x 2 X

m{ 1

’

k = 0

g(f,^kx)

1/m