Sensitivity, Local Stable/Unstable Sets and Shadowing

The uploaded paper proves exactly the claim in Theorem A: on a Peano continuum, if f is sensitive with sensitivity constant ε, then for every x the ε-unstable continuum Cu_ε(x) has diameter at least ε; the analogous statement holds for f^{-1} and stable continua. The statement and the proof outline appear explicitly in the text, including the construction via connected components of small balls around past iterates x_k = f^{-m_k}(x), the definition of a minimal radius δ_k ensuring first expansion beyond ε at time m_k, and the passage to a Hausdorff limit C_x ⊂ W^u_ε(x) with diam(C_x)=ε (see Theorem A and its proof, and the steps detailing D_k, C_k, and the ak/bk argument: ). By contrast, the model’s solution stalls at a key step, seeking to pull back a forward ε-separation to time 0 by showing limsup d(x, w_k) ≥ ε; it acknowledges that this requires an additional uniform distortion argument that is not provided. The paper circumvents this difficulty by constructing continua of diameter ε that already contain x and sit inside W^u_ε(x), eliminating the need for a pointwise pullback lower bound. Hence, the paper’s proof is complete and correct for both (i) and (ii), while the model’s solution is incomplete.