Holomorphic motion of equilibrium states

Joint work with Fabrizio Bianchi.
Let $f:\mathbb{P}^k\to \mathbb{P}^k$ be a holomorphic endomorphism and $v$ a real-valued continuous function on $\mathbb{P}^k$. In this talk we briefly introduce the notion of equilibrium states $\mu_\phi$, a measure mixing measure supported in a small Julia set of $f$ and weighted on $\psi$.
Let us given $f:=(\lambda,f_\lambda),$ $\lambda\in M$ a stable family of endomorphisms and for some $\lambda_0\in M$ an equilibrium state $\mu_{\psi_{\lambda_0}}$ for $f_{\lambda_0}$. We prove the existence an acritical, ergodic $\mu_{\psi_{\lambda_0}}$-web (measure) $\mathcal{M}$ on $\mathcal{J}:=\{\gamma\in\mathcal{O}(M,\mathbb{P}^k): \gamma(\lambda)\in J_\lambda,\forall \lambda\in M\}$ such that its projection at $\lambda_0$ is $\mu_{\psi_{\lambda_0}}$ and its projection at $\lambda$ is again an equilibrium state.