Symbolic dynamics for certain non-invertible $C^{1+β}$ maps

Let $f$ be a non-invertible $C^{1+β}(β>0)$ map with zero Lyapunov exponents and singularities on a closed Riemannian manifold $M$. We consider the symbolic dynamics of $f$. Combining the techniques in recent works of Sarig, Ovadia and Araujo-Lima-Poletti, we construct a countable Markov partition for the invariant set consisting of summable points of the inverse limit space of $(M, f)$ and show that there exists a finite-to-one symbolic extension for $f$ on the corresponding subset of $M$.