Convergence speeds of iterations of Ruelle operator with weakly contractive IFS

Abstract

Given Ruelle operator S g induced by a summable variation potential function g on a symbolic space, Fan and Pollicott (Math. Proc. Cambridge Philos. Soc. 129 (2000) 99–115) gave an estimate of the convergence speed of iterations { S g n f} n=1 ∞. Later on, Fan and Jiang (Comm. Math. Phys. 223 (2001) 143–159) extended it to locally expansive Dini dynamical system. It is known that the systems they considered have the bounded distortion property (BDP), and the BDP is a key condition in the papers. We extend their results to weakly contractive Dini iterated function systems ( X,{ w j } j=1 m ,{ p j } j=1 m ) which may not have the BDP and allows to have overlapping.