Non-uniform hyperbolicity of maps on $\mathbb{T}^2$

In this paper we prove that the homotopy class of non-homothety linear endomorphisms on $\mathbb{T}^2$ with determinant greater than 2 contains a $C^1$ open set of non-uniformly hyperbolic endomorphisms. Furthermore, we prove that the homotopy class of non-hyperbolic elements (having either $1$ or $-1$ as an eigenvalue) whose degree is large enough contains non-uniformly hyperbolic endomorphisms that are also $C^2$ stably ergodic. These results provide partial answers to certain questions posed in arXiv:2206.08295v2