## Abstract

We quantitatively relate the Patterson-Sullivan currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on outer space and to Guirardel's intersection number. Thus we show that, given N ≥ 2 and ε > 0, there exists a constant c = c(N, ε) > 0 such that for any two trees T,S 2 cv_{N} of covolume 1 and injectivity radius ≥ ε, we have |log<S,μ_{T}> - d_{L}(T,S)| ≤ c, where d_{L} is the asymmetric Lipschitz metric on the Culler-Vogtmann outer space, and where μ_{T} is the (appropriately normalized) Patterson-Sullivan current corresponding to T . As a corollary, we show there exist constants C_{1} ≥ 1 and C_{2} ≥ 1 (depending on N, ε) such that for any T,S as above we have 1/C_{1} log i_{c}(T,S)-C_{2} ≤ log < S, μ_{T}> ≤ C_{1} log i_{c} (T,S)+C_{2}, where i_{c} is the combinatorial version of Guirardel's intersection number. We apply these results to the properties of generic stretching factors of free group automorphisms. In particular, we show that for any N ≥ 2, there exists a constant 0 < ϕ_{N} < 1 such that for every automorphism ϕ of F_{N} = F(A), we have 0 < ρ_{N} ≤ λ_{A}(ϕ)/Λ_{A}(ϕ)≤ 1. Here λA is the generic stretching factor of ϕ with respect to the free basis A of F_{N} and λ_{A}(ϕ) is the extremal stretching factor of ϕ with respect to A.

Original language | English (US) |
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Pages (from-to) | 371-398 |

Number of pages | 28 |

Journal | Pacific Journal of Mathematics |

Volume | 277 |

Issue number | 2 |

DOIs | |

State | Published - 2015 |

## Keywords

- Culler-Vogtmann's outer space
- Patterson-Sullivan measures
- geodesic currents

## ASJC Scopus subject areas

- Mathematics(all)