Degeneration of dynamical degrees in families of maps
Volume 184 / 2018
Acta Arithmetica 184 (2018), 101-116
MSC: Primary 37P05; Secondary 37P30, 37P55, 11G50.
DOI: 10.4064/aa8620-5-2017
Published online: 2 July 2018
Abstract
The dynamical degree of a dominant rational map $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ is the quantity $\delta(f):=\lim\,(\deg f^n)^{1/n}$. 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) $\delta(f_t) \lt \delta(f_T)$; (3) $\delta(f_t) \lt \delta(f_T)$ and $\delta(g_t) \lt \delta(g_T)$ for “independent” families of maps. We give a sufficient condition for our conjecture to hold and prove that the condition is true for monomial maps. We describe non-trivial families of maps for which our questions have affirmative and negative answers.
