### Abstract

Let h be the Hausdorff dimension of the Julia set J(R) of a Misiurewicz's rational map R : {Mathematical expression} (subexpanding case). We prove that the h-dimensional Hausdorff measure H_{ h} on J(R) is finite, positive and the only h-conformal measure for R : {Mathematical expression} up to a multiplicative constant. Moreover, we show that there exists a unique R-invariant measure on J(R) equivalent to H_{ h} .

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

Pages (from-to) | 193-214 |

Number of pages | 22 |

Journal | Israel Journal of Mathematics |

Volume | 76 |

Issue number | 1-2 |

DOIs | |

State | Published - Oct 1 1991 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*76*(1-2), 193-214. https://doi.org/10.1007/BF02782852

}

*Israel Journal of Mathematics*, vol. 76, no. 1-2, pp. 193-214. https://doi.org/10.1007/BF02782852

**Hausdorff measures on Julia sets of subexpanding rational maps.** / Denker, M.; Urbański, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Hausdorff measures on Julia sets of subexpanding rational maps

AU - Denker, M.

AU - Urbański, M.

PY - 1991/10/1

Y1 - 1991/10/1

N2 - Let h be the Hausdorff dimension of the Julia set J(R) of a Misiurewicz's rational map R : {Mathematical expression} (subexpanding case). We prove that the h-dimensional Hausdorff measure H h on J(R) is finite, positive and the only h-conformal measure for R : {Mathematical expression} up to a multiplicative constant. Moreover, we show that there exists a unique R-invariant measure on J(R) equivalent to H h .

AB - Let h be the Hausdorff dimension of the Julia set J(R) of a Misiurewicz's rational map R : {Mathematical expression} (subexpanding case). We prove that the h-dimensional Hausdorff measure H h on J(R) is finite, positive and the only h-conformal measure for R : {Mathematical expression} up to a multiplicative constant. Moreover, we show that there exists a unique R-invariant measure on J(R) equivalent to H h .

UR - http://www.scopus.com/inward/record.url?scp=51249176438&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249176438&partnerID=8YFLogxK

U2 - 10.1007/BF02782852

DO - 10.1007/BF02782852

M3 - Article

AN - SCOPUS:51249176438

VL - 76

SP - 193

EP - 214

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1-2

ER -