Deformation and perturbative rigidity near de la Llave examples

De la Llave's examples are Anosov diffeomorphisms on the four-torus $\mathbb{T}^4$ with constant Lyapunov spectrum, yet they are not $C^{1}$-conjugate to the linear model or to each other. Nevertheless, we show that such examples are ``locally exceptional'': we prove deformation and local rigidity for generic diffeomorphisms in proximity of de la Llave's examples.