# Degeneration of Dynamical Degrees in Families of Dominant Maps $\mathbb{P}^N\dashrightarrow\mathbb{P}^N$

The *dynamical degree* $\delta(f)$ of a dominant rational map $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ measures the average growth rate of the degree of the iterates of $f$. It is defined as the limit \[ \delta(f):=\lim_{n\to\infty} (\deg f^n)^{1/n}. \] We consider the variation of dynamical degrees in 1-parameter families of maps $f_T$. We make three conjectures 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 "sufficiently independent" families of maps.

We give a sufficient condition for the first conjecture to hold and show that it is true for monomial maps, and we provide evidence for the second and third conjectures by proving them for certain non-trivial families. (Research supported by Simons Collaboration Grant #241309. Joint work with Greg Call.)