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 cvN of covolume 1 and injectivity radius ≥ ε, we have |log<S,μT> - dL(T,S)| ≤ c, where dL is the asymmetric Lipschitz metric on the Culler-Vogtmann outer space, and where μT is the (appropriately normalized) Patterson-Sullivan current corresponding to T . As a corollary, we show there exist constants C1 ≥ 1 and C2 ≥ 1 (depending on N, ε) such that for any T,S as above we have 1/C1 log ic(T,S)-C2 ≤ log < S, μT> ≤ C1 log ic (T,S)+C2, where ic 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 < ϕN < 1 such that for every automorphism ϕ of FN = F(A), we have 0 < ρN ≤ λA(ϕ)/ΛA(ϕ)≤ 1. Here λA is the generic stretching factor of ϕ with respect to the free basis A of FN and λA(ϕ) is the extremal stretching factor of ϕ with respect to A.
- Culler-Vogtmann's outer space
- Patterson-Sullivan measures
- geodesic currents
ASJC Scopus subject areas