Decay of correlations for certain quadratic maps (Q1193113)

The paper deals with the dynamics of the maps \(f_ a\), \(a\in[0,2]\), of the interval \([-1,1]\), defined by \(f_ a(x)=1-ax^ 2\) for all \(x\in[- 1,1]\). It has been known that for \(a\) in a certain set, say \(\Delta_ 0\), of positive Lebesgue measure \(f_ a\) has an absolutely continuous invariant measure. The author proves that for \(a\) in a certain subset \(\Delta\) of \(\Delta_ 0\) of positive Lebesgue measure such a measure, say \(\mu\), is unique, \((f_ a,\mu)\) is an exact endomorphism and the correlation function (with respect to \(\mu\)) of any two functions of bounded variation decays exponentially, in the sense that there exists a \(\tau\in(0,1)\) such that given functions \(\varphi\) and \(\psi\) of bounded variation there exists a constant \(C=C(\varphi,\psi)\) such that \[ \big| \int\varphi\cdot (\psi\circ f^ n_ a)\;d\mu-\int\varphi\;d\mu\cdot \int\psi\;d\mu\big|\leq C\tau^ n \text{ for all } n\geq 0; \] the author also proves the central limit theorem for the sequence \(\{\varphi\circ f^ n_ a\}\) for \(\varphi\) of bounded variation, for these \(a\). It may be mentioned that the sets \(\Delta_ 0\) and \(\Delta\) as above are described in terms of simple conditions on the values of the iterates of \(f_ a\) and their derivatives. The author comments that the results generalize to other maps of the interval, satisfying what is known as the Misiurewicz condition.