Robust Transitivity and Domination for Endomorphisms Displaying Critical Points

The paper’s Theorem A explicitly proves that every C^1-robustly transitive endomorphism with critical points admits a nontrivial dominated splitting over the inverse-limit space, with a uniform domination time and a uniform angle between the bundles, via a carefully constructed dichotomy (Theorem C) based on kernel-dimension growth, not on creating sources/sinks. It introduces κ from the Key Obstruction and defines the candidate splitting using visits to Cr_κ(f) and the times τ_i^± (equations (2), (∗), (∗∗)), then establishes uniform angle and domination before extending to all orbits (via density and an extension proposition) . The candidate solution diverges materially at two critical points: (i) it defines F(x_i) as an intersection of images Df^n(T_{x_{i-n}}M), claiming the sequence is nested and has dimension ≥ d−κ. For non-invertible maps, these subspaces are not nested; the intersection can be much smaller, and invariance/injectivity is not justified. This contrasts with the paper’s construction E(x_i)=ker(Df^{m_f+τ_i^+}) and F(x_i)=Im(Df^{|τ_i^-|}) tailored to Cr_κ(f) and mf, which ensures the right dimensions and invariance . (ii) It invokes a BDP-type sink/source dichotomy to force domination. In the non-invertible setting with critical points, the paper uses a different dichotomy: lack of domination yields nearby maps with larger kernel dimension at some iterate (hence contradicting robust transitivity via the Key Obstruction), together with a rotation/angle lemma to secure uniform angles, and a delicate control near the critical set (Lemma 4.3) rather than source/sink creation . The model omits the uniform angle requirement present in the paper’s definition of dominated splitting for maps with critical points and does not explain the extension from a dense invariant set to all of Mf (addressed via Proposition 2.10 in the paper) .