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arXiv · 1609.02119

Degeneration of Dynamical Degrees in Families of Maps

Joseph H. Silverman·Gregory Call
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Abstract

The dynamical degree of a dominant rational map $f:\mathbb{P}^N\rightarrow\mathbb{P}^N$ is the quantity $δ(f):=\lim(\text{deg} f^n)^{1/n}$. 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) $δ(f_t)\leδ(f_T)-ε$; (2) $δ(f_t)<δ(f_T)$; (3) $δ(f_t)<δ(f_T)$ and $δ(g_t)<δ(g_T)$ 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.

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BibTeXRIS

Joseph H. Silverman, Gregory Call. 2018-07-30. Degeneration of Dynamical Degrees in Families of Maps. https://doi.org/10.4064/aa8620-5-2017

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