The quadric threefold dynamical-degree and periodic-point conjecture

Let $F:\mathbb{C}^4\rightarrow\mathbb{C}^4$ be the birational map

$$F(x_1,x_2,x_3,x_4)=(x_2,-x_4,x_1-x_1x_2^2,-x_3+x_1x_2x_4),$$

let $\phi(x_1,x_2,x_3,x_4)=x_1x_4-x_2x_3$, and for generic $c\in\mathbb{C}$ let $X_c\subset\mathbb{P}^4$ be the smooth projective quadric closure of $\phi^{-1}(c)$. Write $f_c=F|_{Z_c}$ for $Z_c=\phi^{-1}(c)$ and $\widehat f_c$ for its extension to $X_c$; let $IsoFix_n(f_c)$ denote its isolated fixed points of the relevant iterate. Quadric threefold conjecture. The first dynamical degree $\lambda_1(\widehat f_c)$ is the largest root $\zeta_1$ of $t^3-t^2-t-1$, approximately $1.8393$; the second dynamical degree $\lambda_2(\widehat f_c)$ is the largest root $\zeta_2$ of $2t^3-3(t^2-1)-4t$, approximately $2.1108$, and is algebraic but not an algebraic integer; and

$$\limsup_{n\rightarrow\infty}\frac{\log\sharp(IsoFix_n(f_c))}{n}\leq\log 2.108.$$

The weaker asserted estimate is

$$\limsup_{n\rightarrow\infty}\frac{\log\sharp(IsoFix_{2n+1}(f_c))}{2n+1}\leq\log 2.108.$$

These proposed values and bounds are presented as candidate counterexamples to the expected isolated-periodic-point growth and algebraic-integrality conjectures.