Existence of robust non-uniformly hyperbolic endomorphism in homotopy classes

We extend the results of arXiv:2206.08295v2 by showing that any homothety in ${\mathbb{T}}^2$ is homotopic to a non-uniformly hyperbolic ergodic area preserving map, provided that its degree is at least $5^2$. We also address other small topological degree cases not considered in the previous article. This proves the existence of a ${\mathcal{C}}^1$ open set of non-uniformly hyperbolic systems, that intersects essentially every homotopy classes in ${\mathbb{T}}^2$, where the Lyapunov exponents vary continuously.