### 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) |
---|---|

Pages (from-to) | 833-847 |

Number of pages | 15 |

Journal | Transactions of the American Mathematical Society |

Volume | 315 |

Issue number | 2 |

DOIs | |

State | Published - 1989 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Regularity of the metric entropy for expanding maps.** / Rychlik, Marek R.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 315, no. 2, pp. 833-847. https://doi.org/10.1090/S0002-9947-1989-0958899-6

}

TY - JOUR

T1 - Regularity of the metric entropy for expanding maps

AU - Rychlik, Marek R

PY - 1989

Y1 - 1989

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=0040988958&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040988958&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1989-0958899-6

DO - 10.1090/S0002-9947-1989-0958899-6

M3 - Article

AN - SCOPUS:0040988958

VL - 315

SP - 833

EP - 847

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 2

ER -