Non-invertible case (δ_L>0, δ_R<0): 2^n-component structure in R^{(3)}_n

Determine whether for every integer n ≥ 1 and every parameter ξ in R^{(3)}_n = { ξ ∈ Φ^{(3)} | φ_min(g^n(ξ)) > 0, φ_min(g^{n+1}(ξ)) ≤ 0, α(ξ) < 0 }, the border-collision normal form f_ξ has a chaotic attractor with exactly 2^n connected components.

Background

Building on the renormalisation operator g and the functions φ+, φ−, and φ_min, the authors subdivide the non-invertible parameter regime into regions R{(3)}_n. Proposition 8.1 shows how g maps these regions hierarchically, suggesting a doubling of components.

Their numerical evidence indicates a component-doubling structure, but a formal proof for all n ≥ 1 is not provided. The sentence below explicitly labels this as a conjecture.

References

Proposition \ref{pr:gmaps3} suggests that throughout each R{(3)}_n with n \ge 1 the BCNF has an attractor with exactly 2n connected components, and Fig.~\ref{fig:reg3} supports this conjecture.

— The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps (2402.05393 - Ghosh et al., 8 Feb 2024) in Section 8 (The non-invertible case δ_L > 0, δ_R < 0)