Degeneration of Dynamical Degrees in Families of Maps
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 1parameter 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 nontrivial families of maps for which our questions have affirmative and negative answers.
 Publication:

arXiv eprints
 Pub Date:
 September 2016
 DOI:
 10.48550/arXiv.1609.02119
 arXiv:
 arXiv:1609.02119
 Bibcode:
 2016arXiv160902119S
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Dynamical Systems;
 37P05 (Primary);
 37P30;
 37P55 (Secondary)
 EPrint:
 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)