Degeneration of Dynamical Degrees in Families of Maps

TL;DR

This paper investigates how the dynamical degree varies in families of rational maps, proposing conjectures, providing conditions for their validity, and analyzing specific cases like monomial maps.

Contribution

It introduces a conjecture on the degeneration of dynamical degrees in families of maps and proves it for monomial maps, also exploring various examples and conditions.

Findings

Conjecture on the degeneration of dynamical degrees proposed.

Proved the conjecture for monomial maps.

Identified families with both affirmative and negative answers to the posed questions.

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.