Zero entropy for some birational maps of \(\mathbb{C}^2\) (Q2633671)

The authors consider birational self-maps of \(\mathbb C^2\) of the form \[ f(x,y)=\left(\alpha_0+\alpha_1x+\alpha_2y,\frac{\beta_0+\beta_1x+\beta_2y}{\gamma_0+\gamma_1x+\gamma_2y}\right),\quad(\gamma_1,\gamma_2)\neq(0,0). \] Attention focuses on the cases \(\alpha_1\gamma_2 - \alpha_2\gamma_1 = 0\) and \(\beta_1\gamma_2 - \beta_2\gamma_1 = 0\). The authors study the sequence \((d_n)\) of degrees \(d_n := \deg(f^n)\). They also discuss which of the above maps have zero entropy (in the algebraic sense). Following \textit{J. Diller} and \textit{C. Favre} [Am. J. Math. 123, No. 6, 1135--1169 (2001; Zbl 1112.37308)], in the cases where \((d_n)\) grows sub-exponentially, the authors give the corresponding invariant fibrations responsible for this phenomenon. Further properties studied include integrability and periodicity.