Bifurcation currents in one-dimensional holomorphic dynamics

Theorem 1.3.5 The Lyapunov exponent of a degree d rational map is always greaterthan 1 ln(d) and the equality occurs if and only if the map is a Lattès example.2We recall that a Lattès map is, by definition, induced on the Riemann spherefrom an expanding map on a complex torus by mean of some elliptic function. Werefer to the survey paper of Milnor [Mi2] for a detailed discussion of these maps.A remarkable consequence of the positivity of L(f) is that the iterated inversebranch f −nˆx(see definition 1.2.9) are approximately e −nL -Lipschiptz and are definedon a disc whose size only depends on ˆx.Proposition 1.3.6 There exists ɛ 0 > 0 and, for ɛ ∈]0, ɛ 0 ], two measurable functionsη ɛ : ̂Xreg →]0, 1] and S ɛ : ̂Xreg →]1, +∞] such that the maps f −nˆxare defined onD(x 0 , η ɛ (ˆx)) and Lip f −nˆx≤ S ɛ (ˆx)e −n(L−ɛ) for every n ∈ N and ˆµ-a.e. ˆx ∈ ̂X reg .Proof. We may assume that 0 < ɛ 0 < L −n. Since f3 ˆxassertion of Lemma 1.2.11 yields ln Lip f −nˆxergodic theorem we thus havelim sup 1 nln Lip f−nˆx≤ −L + ɛ 3≤ n ɛ 3 − ∑ n= fx −1−n◦ · · · ◦ fx −1−1, the thirdk=1 ln ρ(x −k). By Birkhofffor ˆµ-a.e. ˆx ∈ ̂X.Then there exists n 0 (ˆx) such(that Lip f −nˆx )≤ e −n(L−ɛ) for n ≥ n 0 (ˆx) and it sufficesto set S ɛ := max 0≤n≤n0 (ˆx) e n(L−ɛ) Lip f −nˆx to get the estimateLip f −nˆx≤ S ɛ (ˆx)e −n(L−ɛ) for every n ∈ N and ˆµ-a.e. ˆx ∈ ̂X reg .We now set η ɛ := αɛS ɛwhere α ɛ is the given by Proposition 1.2.10. Let us check byinduction on n ∈ N that f −nˆxis defined on D(x 0 , η ɛ (ˆx)) for ˆµ-a.e. ˆx ∈ ̂X and everyn ∈ N. Here we will use the fact that the function α ɛ is slow: α ɛ (τ(ˆx)) ≥ e −ɛ α ɛ (ˆx).Assume that f −nˆxis defined on D ( x 0 , η ɛ (ˆx) ) . Then, by our estimate on Lip f −nˆx, wehave( (f −nˆx D x0 , η ɛ (ˆx) )) ⊂ D ( x −n , e −n(L−ɛ) α ɛ (ˆx) ) .On the other hand, by Proposition 1.2.10, the branch fx −1−n−1is defined on the discD ( x −n , α ɛ (τ n+1 (ˆx)) ) which, as α ɛ is slow, contains D ( x −n , e −(n+1)ɛ α ɛ (ˆx) ) . Now,since 0 < ɛ 0 < L one has 3e−(n+1)ɛ ≥ e −n(L−ɛ) and thus f −(n+1)ˆxdefined on D ( x 0 , η ɛ (ˆx) ) .= f −1x −n−1◦ f −nˆxis⊓⊔1.3.2 Lyapunov exponent and multipliers of repelling cyclesThe following approximation property will play an important role in our study ofbifurcation currents. We would like to mention that Deroin and Dujardin haverecently used similar ideas to study the bifurcation in the context of kleinean groups(see [DD]).18