### 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 C^{2}-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 f_{μ}(x) = 1 − μ^{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).

Original language | English (US) |
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Pages (from-to) | 833-847 |

Number of pages | 15 |

Journal | Transactions of the American Mathematical Society |

Volume | 315 |

Issue number | 2 |

DOIs | |

State | Published - Oct 1989 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics