A SUBSHIFT WITHOUT DOUBLY-ASYMPTOTIC POINTS

The paper’s Theorem 1 states precisely the claim: for the 3-interval exchange T with lengths (a, b−a, 1−b) and 0<a<b<1 rationally independent, if α is the itinerary of a regular x0, then X_α is infinite and the subshift σ: X_α→X_α has no doubly-asymptotic points. The paper proves this via injectivity of the itinerary map and a case analysis of limit points near 1, a±, b± (see the theorem statement and proof passages) , building on the preliminaries on IETs and the Keane condition/minimality . The candidate solution is also correct but follows a different route: it uses translation-sum invariants S(w), shrinking cylinder intersections to obtain a unique realization map π, and rational independence to exclude S_r=0, thereby forbidding distinct doubly-asymptotic pairs. No contradictions were found; thus both are correct and essentially prove the same theorem by different methods.