PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Degeneration of dynamical degrees in families of maps

Volume 184 / 2018

Joseph H. Silverman, Gregory S. Call Acta Arithmetica 184 (2018), 101-116 MSC: Primary 37P05; Secondary 37P30, 37P55, 11G50. DOI: 10.4064/aa8620-5-2017 Published online: 2 July 2018


The dynamical degree of a dominant rational map $f:\mathbb{P}^N\dashrightarrow\mathbb{P}^N$ is the quantity $\delta(f):=\lim\,(\deg f^n)^{1/n}$. 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) $\delta(f_t) \lt \delta(f_T)$; (3) $\delta(f_t) \lt \delta(f_T)$ and $\delta(g_t) \lt \delta(g_T)$ for “independent” families of maps. We give a sufficient condition for our conjecture to hold and prove that the condition is true for monomial maps. We describe non-trivial families of maps for which our questions have affirmative and negative answers.


  • Joseph H. SilvermanMathematics Department
    Box 1917 Brown University
    Providence, RI 02912, U.S.A.
  • Gregory S. CallDepartment of Mathematics and Statistics
    Amherst College
    Amherst, MA 01002, U.S.A.

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image