Self-projective sets

The paper defines Λ_A via rank-one elements in R_G and proves Theorem 1.2: for irreducible G containing a proximal element, Λ_A is exactly the closure of attracting fixed points of proximal elements. The paper’s proof uses a cone argument and the Krein–Rutman theorem on sequences of the form A_n A A_n to obtain proximal elements with fixed points arbitrarily close to a prescribed point in Λ_A, after first showing the easy inclusion via λ−n A^n → π of rank one. These steps are explicitly stated in the survey (definition and theorem statement) and sketched in the proof of Theorem 1.2. The candidate solution proves the same theorem with a different perturbative argument: it perturbs a rank–one ρ ∈ R_G with tr ρ ≠ 0 and shows matrices sufficiently close to ρ are proximal with attracting lines close to P(Im ρ), then approximates ρ by k_n g_n with g_n ∈ G. This gives the desired density and completes both inclusions. Both arguments are coherent under the same hypotheses; they differ methodologically but reach the same conclusion. See Definition 1.1 and Theorem 1.2 for the statement, and the proof outline invoking λ−n A^n and the Krein–Rutman step for density in the paper’s treatment.