Degeneration of Dynamical Degrees in Families of Maps

The dynamical degree of a dominant rational map $f:mathbb{P}^Nrightarrowmathbb{P}^N$ 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) $delta(f_t)ledelta(f_T)-epsilon$; (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.