The Graph of the Logistic Map Is a Tower

The paper proves that the chain-recurrence graph of the logistic map is a tower (between any two distinct nodes there is exactly one directed edge, oriented from the node farther from the critical point to the closer one) and that the tower is infinite if and only if the parameter lies in the almost periodic set AAP; see the definition of the graph and towers and the no-loops property, then Theorem 3.2 establishing the tower structure, together with the ρ-ordering via Proposition 3.1 and the cyclic-trapping-region machinery that ensures the direction of edges . The candidate’s core lemma—"the precritical set is dense, hence cylinder meshes shrink to zero, hence from any open set U one can map onto any target y"—is false (e.g., at μ=2 the only preimage of c=1/2 is c itself, so precritical points are not dense), invalidating the covering argument and the constructed two-sided trajectory. Their conclusion that there is an edge from every node to every other node would also contradict the Conley graph’s acyclicity (no edges in both directions) noted by the paper . The paper’s statement that the tower is infinite exactly for parameters with a Cantor (adding-machine) attractor matches its AAP classification , whereas the candidate’s counting argument appeals to external results but is not tied into a correct edge construction.