Ergodic behaviour of higher order derivatives of certain Perron-Frobenius operator
by
Adriana Berechet
Centre for Mathematical Statistics, Bucharest, Romania

We consider a measurable fibred system (B, \Sigma, T) in a class defined by assumptions near that from [7] and [1]; then an unique endomorphism (T, \mu) exists and (B, \Sigma, T, \mu) is a measurable dynamical system. The assumptions induce a partition of B with a finite number of cells.

We consider the Banach spaces L^s, and C^s, s in \bbb N_* containing all functions defined on B having continuous and respectively Lipschitz continuous derivative of order s on every cell and the Perron-Frobenius operator U associated with T with respect to the Lebesgue measure \lambda.

We assume that a condition we call (E_r), where r in \bbb N_* on derivatives of orders s <= r+1 on Jacobians of T^-n, n 1is satisfied. We begin by finding a superior bound of derivativeson cells of orders s r of U^n, n 1.The main theorem asserts the aperiodicity of U consideredon each of spaces L^s and C^s+1, s r.The relation d d C^r+1

Date received: June 24, 2002