### Abstract

In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved.

Original language | English (US) |
---|---|

Pages (from-to) | 217-236 |

Number of pages | 20 |

Journal | Communications in Mathematical Physics |

Volume | 150 |

Issue number | 2 |

DOIs | |

State | Published - Nov 1992 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

**Regularity and other properties of absolutely continuous invariant measures for the quadratic family.** / Rychlik, Marek R; Sorets, Eugene.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 150, no. 2, pp. 217-236. https://doi.org/10.1007/BF02096659

}

TY - JOUR

T1 - Regularity and other properties of absolutely continuous invariant measures for the quadratic family

AU - Rychlik, Marek R

AU - Sorets, Eugene

PY - 1992/11

Y1 - 1992/11

N2 - In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved.

AB - In the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of Jakobson's Theorem. In particular, we show that the density of the invariant measure is continuous at Misiurewicz points. From this we deduce that the Lyapunov exponent is also continuous at these points (our considerations apply just to the parameters constructed in the proof of Jakobson's Theorem). Other properties, like the positivity of the Lyapunov exponent, uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system, are also proved.

UR - http://www.scopus.com/inward/record.url?scp=21144472608&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=21144472608&partnerID=8YFLogxK

U2 - 10.1007/BF02096659

DO - 10.1007/BF02096659

M3 - Article

AN - SCOPUS:21144472608

VL - 150

SP - 217

EP - 236

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -