## Abstract

Let μ be a measure on SL_{2}(R) generating a non-compact and totally irreducible subgroup, and let ν be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if μ is supported on finitely many matrices with algebraic entries, then (Formula Presented.)where h_{RW}(μ) is the random walk entropy of μ, is the Lyapunov exponent for the random matrix product associated with μ, and dim denotes pointwise dimension. In particular, for every δ> 0 , there is a neighborhood U of the identity in SL_{2}(R) such that if a measure μ∈ P(U) is supported on algebraic matrices with all atoms of size at least δ, and generates a group which is non-compact and totally irreducible, then its stationary measure ν satisfies dim ν= 1.

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

Pages (from-to) | 815-875 |

Number of pages | 61 |

Journal | Inventiones Mathematicae |

Volume | 210 |

Issue number | 3 |

DOIs | |

State | Published - 1 Dec 2017 |

### Bibliographical note

Publisher Copyright:© 2017, Springer-Verlag GmbH Germany.

## Keywords

- 37F35

## Fingerprint

Dive into the research topics of 'On the dimension of Furstenberg measure for SL_{2}(R) random matrix products'. Together they form a unique fingerprint.