Dynamical systems with bounded condition and C∗-algebras

The paper proves that under the bounded condition there is an injective, order-preserving map K ↦ H_K from f-invariant sets to reducing subspaces, and that under the totally uniqueness condition this map is bijective (Theorems 3.6 and 3.7). Its surjectivity proof uses the projections P_m = T_{i_1}^*⋯T_{i_m}^* T_{i_m}⋯T_{i_1} built from the itinerary of x, showing P_m a → ⟨a,e_x⟩ e_x and then defining K = {x : ⟨a, e_x⟩ ≠ 0 for some a ∈ M} to get M = H_K . The candidate solution proves the same classification but via a different operator-theoretic route: it identifies the same P_m as cylinder projections E_w = T_w^* T_w, proves E_{w_ℓ} ↓ p_x strongly using total uniqueness, and then uses commutation P_M E_{w_ℓ} = E_{w_ℓ} P_M and strong limits to diagonalize P_M and conclude M = H_K. Both arguments rely on the same hypotheses (bounded condition to define the Ti, total uniqueness for the rank-one/point mass limit) and reach the same conclusion; the model’s proof is a clean SOT-limiting variant of the paper’s constructive argument. The definitions of the setting and of the totally uniqueness condition match the paper’s setup .