Degeneration of Dynamical Degrees in Families of Maps

Abstract

The dynamical degree of a dominant rational map $f:{\mathbb{P}}^N\to {\mathbb{P}}^N$ 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.

18 pages. This is an expanded version of the article publishd in Acta Arithmetica. It contains a corrected statement and full proof of Propostion 11(c)

References in corpus (1)

• A classification of degree $2$ semi-stable rational maps ${\mathbb{P}}^2\to {\mathbb{P}}^2$ with large finite dynamical automorphism group

Cited by in corpus (1)

• Non-density of points of small arithmetic degrees