MAXIMAL EQUICONTINUOUS FACTOR AND MINIMAL MAP ON FINITELY SUSLINEAN CONTINUA

The paper proves that any minimal map on a finitely Suslinian continuum is conjugate to an irrational circle rotation, via: (i) monotonicity of the maximal equicontinuous factor (MEF) for onto maps on locally connected continua, (ii) non–weak mixing on finitely Suslinian continua, (iii) identification of the equicontinuous factor as a circle rotation, and (iv) a lemma forcing the factor map to be a homeomorphism once all fibres have empty interior. These steps are present and coherent in the PDF. The candidate solution mirrors the paper but contains a critical gap in Step 5: it claims that the existence of a nontrivial asymptotic pair would force periodicity on the equicontinuous factor. In general, an asymptotic pair in the extension only implies the pair lies in the same fibre; it does not force periodicity of a minimal isometry. The correct argument uses minimality to obtain a return of a point to the interior of a fibre, yielding a periodic point on the factor and hence a contradiction with irrational rotation. Thus the paper’s proof is correct; the model’s proof is not complete as written.