Semi-conjugacy rigidity for endomorphisms derived from Anosov on the 2-torus

Let $f$ be a non-invertible partially hyperbolic endomorphism on ${\text{TT}}^2$ which is derived from a non-expanding Anosov endomorphism. Differing from the case of diffeomorphisms derived from Anosov automorphisms, there is no a priori semi-conjugacy between $f$ and its linearization on ${\text{TT}}^2$. We show that $f$ is semi-conjugate to its linearization if and only if $f$ admits a partially hyperbolic splitting with two $Df$-invariant subbundles. Moreover, if we assume that $f$ has an unstable subbundle, then the semi-conjugacy is exactly a topological conjugacy, and the center Lyapunov exponents of the periodic points of $f$ coincide with its linearization. In particular, $f$ is an Anosov endomorphism and the conjugacy is smooth along the stable foliation. For the case that $f$ has a stable subbundle, there is still some rigidity in its stable Lyapunov exponents. However, we also give examples which admit a partially hyperbolic splitting with center subbundle but the semi-conjugacy is indeed non-injective. Finally, we present some applications under the volume-preserving assumption.