Regularity of the metric entropy for expanding maps

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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 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).

Original languageEnglish (US)
Pages (from-to)833-847
Number of pages15
JournalTransactions of the American Mathematical Society
Volume315
Issue number2
DOIs
StatePublished - Oct 1989
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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