Periodicities, rational and elliptic fibrations in some birational maps of $\mathbb{C}^2$ [ Back ]

15:30 to 16:30
CRM - Auditori (C1/034)
Sundus Zafar
Universitat Autònoma de Barcelona


For complex numbers $\alpha_i,\beta_i$ and $\gamma_j, i \in\{0, 1, 2\}, j \in \{0, 2\},$ consider the following family of birational maps:
\[ f (x, y) = (\beta_0 + \beta_1x + \beta_2 y, \frac{\alpha_0 + \alpha_1x + \alpha_2 y}{\gamma_0 + \gamma_2 y}).\]
A classification of this family $f$ into its subfamilies which have bounded growth (in particular periodic growth), linear growth, quadratic growth and exponential growth rate will be discussed. This is done by finding the dynamical degree of the map $f$ and separating the families which have dynamical degree one from the ones which have greater. In the subfamilies with dynamical degree one, two transverse fibrations are found for the families with bounded growth rate. In the periodic case the periodicity of the families is studied and the period of the families is indicated. It is observed that there exist infinite periodic subfamilies of $f,$ depending on the parameter region. The families with linear growth rate are observed to preserve rational fibration. The quadratic growth rate families preserve elliptic fibration that is unique depending on the parameters. In all the cases with dynamical degree one the mappings are found up to affine conjugacy. The subfamilies with dynamical degree greater than one are not included in this session.

This is joint work with Anna Cima.