A Note on the Asymptotics of Perturbed Expanding Maps

Mark Pollicott

Abstract: Given any analytic expanding map $f: M \to M$ on a compact manifold $M$, it is well-known that $f:M \to M$ is exponentially mixing with respect to the smooth invariant measure $\mu$. Our first result is that although for linear expanding maps on tori the rate of mixing is arbitrarily fast, generically this is \Sle{not} the case.\Prgrf For random compositions of $\epsilon$-close analytic expanding maps $g:M\to M$ which also preserve $\mu$ and we show that the rate of mixing for the composition has an upper bound which can be made arbitrarily close to that for the single transformation $f$ by choosing $\epsilon>0$ sufficiently small.

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