Regularity of the metric entropy for expanding maps

Abstract:

The main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the Perron-Frobenius operator for expanding mappings. As a consequence, we are able to show that the metric entropy of an expanding map has modulus of continuity $x\log (1/x)$ on the space of ${C^2}$-expandings. We also give an explicit estimate of the rate of mixing for ${C^1}$-functions in terms of natural constants. It seems that the method we present can be generalized to other classes of dynamical systems, which have a distinguished invariant measure, like $\operatorname {Axiom} \text {A}$ diffeomorphisms. It also can be adopted to show that the entropy of the quadratic family ${f_\mu }(x) = 1 - \mu {x^2}$ computed with respect to the absolutely continuous invariant measure found in Jakobson’s Theorem varies continuously (the last result is going to appear somewhere else).