Ergodic Theory and Dynamical Systems



Decay of correlations for piecewise smooth maps with indifferent fixed points


HUYI HU a1
a1 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA (e-mail: hu@math.psu.edu)

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Abstract

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)