Monotonicity of the Over-Rotation Intervals for Bimodal Maps

The paper proves that for the truncation family H_{α,β} of the bimodal horseshoe H2, the map ψ(α,β)=ρ(I_{α,β}) has connected level sets, i.e., ψ is monotone on the parameter rectangle P; see the definition of the family and ψ, and the statement of the main objective in the introduction, as well as the concluding Corollary 3.19 asserting monotonicity . The paper’s proof proceeds via constructing the leading set Z_{p/q} ‘staircases,’ partitioning P into components G_{p/q} and U_{p/q}, and showing I_{p/q} is a deformation retract of G_{p/q}; continuity of ψ then yields the irrational case . The candidate’s solution essentially invokes this very theorem for bimodal truncations and applies it to H_{α,β}, so the conclusion matches the paper’s main result. While the candidate references kneading monotonicity for context, the decisive step is precisely the paper’s connectedness theorem for this parameterization, hence the two solutions are substantively the same in content/conclusion.