## Degeneration of dynamical degrees in families of maps

### Volume 184 / 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.