Patterson-sullivan currents, generic stretching factors and the asymmetric lipschitz metric for outer space

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 C₁ ≥ 1 and C₂ ≥ 1 (depending on N, ε) such that for any T,S as above we have 1/C₁ log i_c(T,S)-C₂ ≤ log < S, μ_T> ≤ C₁ log i_c (T,S)+C₂, 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.