| Contents |
| Given complex numbers $\alpha_i, \gamma_i$ and $\delta_i,
i=0,...,2$, consider the family of birational maps $f:{\textbf{C}^2} \to
{\textbf{C}^2}$ of the following form
\begin{equation}\label{eq1}
f(x,y) = \left(\alpha _0 + {\alpha _1}x + {\alpha
_2}y,\frac{{{\gamma _0} + {\gamma _1}x + {\gamma _2}y}}{{{\delta _0}
+ {\delta _1}x + {\delta _2}y}}\right).
\end{equation}
This family \ref{eq1} is dynamically classified completely in \cite{CZ}. For all the values of parameters for which the determinants $(\gamma \delta )_{12}$ and $(\alpha \delta )_{12}$ are zero, it is called a \textit{degenerate case}. In general the family (\ref{eq1}) has dynamical degree $D = 2.$ The main interest is to identify the possible subcases of (\ref{eq1}) for all the parameter values. By the help of the associated characteristic polynomial of each subcase/subfamily it is possible to know their growth rate. Therefore finding the dynamical degree $D$ for all the subcases helps to locate the subfamilies with entropy \textit{zero} and the ones where $1 < D < 2$. The subfamilies with zero entropy have rather simpler dynamics than the other subfamilies which have non zero entropy. This talk will focus on providing information of all the existing subcases/subfamilies of $(\ref{eq1})$ when it is degenerate. Then the dynamics of the families with zero entropy will be discussed. The family \ref{eq1} includes a subfamily dynamically studied in \cite{BK2} and also this work provides examples for the theoretical results stated in \cite{DF, FS}.
\begin{thebibliography}{9}
\bibitem{CZ}Cima, A. and Zafar, S. \emph{Classification of a family of birational surface maps via dynamical degree}; Preprint
\bibitem{BK2}Bedford, E. and Kim, K. \emph{Periodicities in Linear Fractional Recurrences: Degree Growth of Birational Surface Maps} Michigan Math. J. {\bf 54} (2006), 647-670.
\bibitem{DF}Diller, J. and Favre, C. \emph{Dynamics of bimeromorphic maps of surfaces} Amer. J. of Math., {\bf 123} (2001), 1135-1169.
\bibitem{FS}Fornaes, J-E and Sibony, N. \emph{Complex dynamics in higher dimension. II} Modern methods in complex analysis, Ann. of Math. Stud. 137, Princeton Univ. Press, 1995, pp. 135-182.
\end{thebibliography} |
|