Ergodic Theory and Dynamical Systems

Decay of correlations for piecewise smooth maps with indifferent fixed points

a1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (e-mail:

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We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0<\gamma < 1$. 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 $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions. We also show that the density function h of $\mu$ has the order $x^{-\gamma}$ as $x\to 0$. Perron–Frobenius operators are the main tool used for proofs.

(Received October 12 2001)
(Revised November 20 2002)