The bifurcation measure $\mu_{f,a}$ of a one-dimensional dynamical pair, i.e. a holomorphic family of a rational map $f=(f_\lambda)_{\lambda\in\Lambda}$ parametrized by a holomorphic curve endowed with a homomorphically moving point $a:\Lambda\to\mathbb{P}^1$, is a positive measure which support is exactly the locus of non-normality of the sequence of iterates of $a$, seen as a family of holomorphic functions $\Lambda\to\mathbb{P}^1$.

In a past joint work with Henry De Thélin and Gabriel Vigny, we defined the notion of entropy of the pair $(f,a)$ proved that the measure $\mu_{f,a}$ has maximal entropy. In a more recent work, we proved that the "parametric Lyapunov exponent" i.e. the $\liminf_n \frac{1}{n}\log|(f^n_\lambda)'(a(\lambda))| $ is $\mu_{f,a}$-almost everywhere greater than or equal to $\log d/D\geq \log d/2$, where $D$ is the lower box-dimension of the measure $\mu_{f,a}$. This generalizes partially a result of Graczyk and Swiatek who proved that for a typical parameter for the harmonic measure of the Mandelbrot set, this Lyapunov exponent is $\log d$.

Our strategy of proof is based on properties of laminar currents and a reverse dynamics-parameter transfer.