### Abstract

We study parabolic iterated function systems (IPS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no *overlaps.* We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

Original language | English |
---|---|

Pages (from-to) | 5145-5164 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 353 |

Issue number | 12 |

Publication status | Published - 2001 |

### Fingerprint

### Keywords

- Iterated function systems
- Parabolic maps
- Random continued fractions

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Transactions of the American Mathematical Society*,

*353*(12), 5145-5164.

**Invariant measures for parabolic ifs with overlaps and random continued fractions.** / Simon, K.; Solomyak, B.; Urbanski, M.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 353, no. 12, pp. 5145-5164.

}

TY - JOUR

T1 - Invariant measures for parabolic ifs with overlaps and random continued fractions

AU - Simon, K.

AU - Solomyak, B.

AU - Urbanski, M.

PY - 2001

Y1 - 2001

N2 - We study parabolic iterated function systems (IPS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no *overlaps.* We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

AB - We study parabolic iterated function systems (IPS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no *overlaps.* We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

KW - Iterated function systems

KW - Parabolic maps

KW - Random continued fractions

UR - http://www.scopus.com/inward/record.url?scp=23044529985&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23044529985&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:23044529985

VL - 353

SP - 5145

EP - 5164

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 12

ER -