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

Abstract

Let f be a non-invertible partially hyperbolic endomorphism on

that 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

T 2 $\mathbb {T}^2$ double struck upper T squared

. We show that f is semi-conjugate to its linearization if and only if f admits a partially hyperbolic splitting with two

D f $Df$ upper D f

-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 where f has a stable subbundle, there is still some rigidity in its stable Lyapunov exponents. However, we also give examples that 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.