Degeneration of Dynamical Degrees in Families of Maps

Abstract

The dynamical degree of a dominant rational map $f:\mathbb{P}^N\rightarrow\mathbb{P}^N$ is the quantity $\delta(f):=\lim(\text{deg} f^n)^{1/n}$. We study the variation of dynamical degrees in 1-parameter families of maps $f_T$. We make a conjecture and ask two questions concerning, respectively, the set of $t$ such that: (1) $\delta(f_t)\le\delta(f_T)-\epsilon$; (2) $\delta(f_t)<\delta(f_T)$; (3) $\delta(f_t)<\delta(f_T)$ and $\delta(g_t)<\delta(g_T)$ for "independent" families of maps. We give a sufficient condition for our conjecture to hold and prove that it is true for monomial maps. We describe non-trivial families of maps for which our questions have affirmative and negative answers.