Degenerations of complex dynamical systems (Q2879419)

This paper considers families of maps in the space \(\mathrm{Rat}_d\) of degree-\(d\) endomorphisms of \(\hat{\mathbb{C}}\); the families of interest are those which have a limit point in \(\partial \mathrm{Rat}_d\), for the natural compactification \(\overline{\mathrm{Rat}}_d = \mathbb{P}^{2d+1}(\mathbb{C})\). The major accomplishment of the paper is the result (Theorem B) that a family \(\{f_t | t \in \mathbb{D}-\{0\}\} \subseteq \mathrm{Rat}_d\) with \(\lim_{t \rightarrow 0} f_t \in \partial \mathrm{Rat}_d\) always yields a probability measure \(\mu_0\) on \(\hat{\mathbb{C}}\) which is the weak limit of the measures of maximal entropy for the functions \(f_t\) as \(t \rightarrow 0\). The paper also shows the result (Theorem A) that \(\mu_0\) is always atomic with countably infinite support.NEWLINENEWLINEThe methodology used to prove Theorem B takes the view that the family of dynamical systems \(f_t:\hat{\mathbb{C}} \rightarrow \hat{\mathbb{C}}\) can be interpreted as a dynamical system on the Berkovich projective line over the completion of the field of formal Puiseux series in \(t\). As noted in the paper, this approach has been used in previous work to prove a variety of results. The crucial additions to this viewpoint here (to which much of the paper is devoted) are the introduction of ``quantized'' measures on Berkovich projective lines and the definition of pull-back actions on spaces of quantized measures. A quantized measure is a measure defined on a Berkovich projective line for a \(\sigma\)-algebra chosen to distinguish some finite set of vertices in the tree structure for the line. The measure \(\mu_0\) is a push-forward of an equilibrium quantized measure for the dynamical system on the Berkovich projective line.NEWLINENEWLINEThe proof of the fact that \(\mu_0\) is atomic with countably infinite support does not require the use of Berkovich projective lines. The key idea for this part of the paper is a clever method of pairing a given sequence of functions \(f_k\) in \(\mathrm{Rat}_d\) with a new sequence of functions composed with Mobius transformations \(A_k \circ f_k\) that has a better convergence behavior in \(\overline{\mathrm{Rat}}_d\).