Decay of correlations for piecewise smooth maps with indifferent fixed points

Abstract

We consider a piecewise smooth expanding map f on the unit interval that has the form f (x) = x + x1+γ+o(x1+γ) near 0, where 0 < γ < 1. We prove by showing both lower and upper bounds that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure μ is polynomial with the same degree 1/γ - 1 for Lipschitz functions. We also show that the density function h of μ has the order x-γ as x → 0. Perron-Frobenius operators are the main tool used for proofs.