TITLE: Weakly hyperbolic invariant tori for two dimensional quasiperiodically forced maps in a degenerate case AUTHORS: Tingting Zhang^(1,3), Angel Jorba^(2), Jianguo Si^(3) (1) College of Mathematics, Qingdao University, Qingdao, Shandong 266071, People's Republic of China (2) Departament de Matematiques i Informatica Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain (3) School of Mathematics, Shandong University, Jinan, Shandong 250100, People's Republic of China E-mails: ting_little@163.com, angel@maia.ub.es, sijgmath@sdu.edu.cn ABSTRACT: In this work we consider a class of degenerate analytic maps of the form \begin{eqnarray*} \left\{ \begin{array}{l} \bar{x} =x+y^{m}+\epsilon f_1(x,y,\theta,\epsilon)+h_1(x,y,\theta,\epsilon),\\ \bar{y}=y+x^{n}+\epsilon f_2(x,y,\theta,\epsilon)+h_2(x,y,\theta,\epsilon),\\ \bar{\theta}=\theta+\omega, \end{array} \right. \end{eqnarray*} where $mn>1,n\geq m,$ $h_1$ and $h_2$ are of order $n+1$ in $z,$ and $\omega=(\omega_1,\omega_2,\ldots,\omega_{d})\in \mathbb{R}^{d}$ is a vector of rationally independent frequencies. It is shown that, under a generic non-degeneracy condition on $f$, if $\omega$ is Diophantine and $\epsilon>0$ is small enough, the map has at least one weakly hyperbolic invariant torus.