We carry out the orbit computation for βφ = (3+√5) /2 ≈ 2. 618, the Pisot number in (2, 3] with minimal polynomial x² − 3x + 1, using the corrected framework of the Problem 1 paper (v6) 5: the Ito–Sadahiro interval Iβφ = −βφ/ (βφ + 1), 1/ (βφ + 1) ) and the shifted greedy map T−βφ (x) = −βφx − ⌊−βφx − ℓβφ⌋. Under this corrected framework: The map is Iβφ-invariant by construction (no exit, no invariance gap). The left endpoint orbit is (2, 1) ^ω: period 2, entering immediately. The right endpoint orbit is (0, 2, 1) ^ω: transient digit 0, then period 2. The admissibility blocks are the same as those of β1 = 1 + √2. This constitutes a third verified instance of D (−β) for β > 2, and the first example of two distinct Pisot numbers in (2, 3 sharing the same admissibility blocks. The contrast with β2 = 1+√3 (blocks (2, 0^ω) and (0, 2, 0^ω) ) sharpens the block-classification question: which property of the minimal polynomial determines the blocks? Note on versions. Versions 1 and 2 of this paper were built on the incorrect framework of P1 v5: the extended interval [−2/ (βφ + 1), 1/ (βφ + 1) ) and the unshifted map T−βφ (x) = −βφx − ⌊−βφx⌋. Under that incorrect framework, the image of the left endpoint appeared to exit the interval at step 1, giving rise to what v2 called the “invariance gap. ” This exit was an artefact of the wrong interval and wrong map. Under the corrected framework the orbit stays in Iβφ throughout, and all content of v2 concerning the invariance gap, the exit strip, Route B as a gap-resolution strategy, and the uniformadmissibility conjecture has been retracted.
Donald G. Palmer (Thu,) studied this question.