Zeta functions and transfer operators for piecewise monotone transformations

Abstract

Given a piecewise monotone transformation T of the interval and a piecewise continuous complex weight function g of bounded variation, we prove that the Ruelle zeta function ζ(z) of (T,g) extends meromorphically to {\z\<θ~x} (where 0=lim H^Γ"" 1 -••• -g°T-g\\ H") and that z is a pole of n-* oo ζ if and only if z" 1 is an eigenvalue of the corresponding transfer operator if. We do not assume that if leaves a reference measure invariant.

Fredholm determinant for higher dimensional piecewise linear transformations

“…Our aim in this article is to prove (cf. [1], [4], [6], [12] Figure 1 which is the same figure as in [12]. Note that Figure 2 On the other hand, if y belongs to outside of ƒÎ, …”

Central Limit Theorem of Mixed Type for Transformations With Quasi-Compact Perron-Frobenius Operators

“…m. So we claim that the measure-theoretic dynamical system (T , μ (i) ) is ergodic for each i. If not, we find an i (say, i = 1 for convenience) and A ∈ B satisfying T −1 A = A m-a.e., A ⊂ (1) and 0 < μ (1) (A) < 1. Then we see from (7) of Proposition A.1 that μ (1) (1) (I (1) − I A ) are T -invariant probability densities.…”

“…If not, we find an i (say, i = 1 for convenience) and A ∈ B satisfying T −1 A = A m-a.e., A ⊂ (1) and 0 < μ (1) (A) < 1. Then we see from (7) of Proposition A.1 that μ (1) (1) (I (1) − I A ) are T -invariant probability densities. Therefore {μ (1) (1) (I (1) − I A ), .…”

Spectrum and Statistical Properties of Chaotic Dynamics

“…In the case of interval dynamics, Keller and I [14] showed that the dynamical zeta function of a piecewise monotone interval map f with a generating partition, and a continuous weight g of bounded variation is analytic in the disc of radius exp(−P (log |g|)) and admits a meromorphic extension to a disc of inverse radius lim sup n→∞ sup |g (n) | 1/n . If g is positive, the poles of ζ g (z) in this disc are related to the poles of the Fourier transform of the correlation function of the equilibrium measure of log g and observables of bounded variation.…”