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 xlog(1/x) on the space of C2-expandings. We also give an explicit estimate of the rate of mixing for C1-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 Axiom A diffeomorphisms. It also can be adopted to show that the entropy of the quadratic family fμ(x) = 1 − μ2computed with respect to the absolutely continuous invariant measure found in Jakobson’s Theorem varies continuously (the last result is going to appear somewhere else).
ASJC Scopus subject areas
- Applied Mathematics