The authors study the bifurcations of a family of holomorphic dynamical systems. In particular, they present a new point of view on bifurcations for the family of quadratic polynomials, which naturally allows them to interpret the set of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three.
The central object of interest is an ergodic dynamical system \((p_c,J_c,\mu_c)_{c\in\mathbb{C}}\), where \(p_c(z)=z^2+c\), \(\mu_c\) is the equilibrium measure of the set \(K_c\), i.e., the set of points with bounded orbits for \(p_c\), and \(J_c=\text{supp} \mu_c\). In order to study the bifurcations it is more natural to consider the holomorphic dynamical system of \(F:\mathbb{C}^2\rightarrow\mathbb{C}^2\) given by \(F(c,z)=(c,p_c(z))\) instead of the family \((p_c)_{c\in\mathbb{C}}\). Such approach has already been used by several authors (see the references in the paper). The novelty in the present paper is that the authors consider the dynamical system that is induced by \(\hat{F}:X\rightarrow X\), where \(X\) is a projectivization of a tangent bundle \(T\mathbb{P}^2\) and \(\hat{F}\) is a rational map which is obtained by lifting \(F\) on \(X\). Since the dynamics of the polynomials \(p_c\) and the bifurcations can be simultaneously observed within the system \(\hat{F}\), the authors propose to study equidistribution properties of \(\hat{F}\) in order to obtain information about the distribution of special values of the parameter \(c\) and its link with the dynamics of \(p_c\).
Let \(m\) be an equilibrium measure of the Mandelbrot set, let \( \mu:=\int\mu_cdm(c) \) be a probability measure on \(\mathbb{P}^2\) and let \(\mathcal{R}=\Pi^*(\mu)\) be a closed positive \((2,2)\)-current on \(X\), where \(\Pi=X\rightarrow \mathbb{P}^2\) is the canonical projection. The main result of this paper is Theorem 1.1. which states that for any smooth closed \((2,2)\)-form \(\Omega\) on \(X\) there exists a constant \(\lambda_{\Omega}\) such that \[ \lim_{n\rightarrow \infty}\frac{1}{n2^n}(\hat{F}^n)^*(\Omega)=\lambda_{\Omega}\mathcal{R}. \] Moreover, the constant \(\lambda_{\Omega}\) only depends on the class of \(\Omega\) in the Hodge cohomology group \(H^{2,2}(X,\mathbb{C})\).
In the proof of Theorem 1.1. the authors first use cohomological arguments to obtain an explicit \((2,2)\)-form for \(\Omega\), which enables them to show that any limit \(T\) of \( \frac{1}{n2^n}(\hat{F}^n)^*(\Omega)\) is of the form \(T=\Pi^*(\nu)\) for some positive measure \(\nu\) on \(\mathbb{P}^2\). The rest of the proof is devoted to prove the equality \(\mu=\nu\).