bibkey: bugeaudkriegershallit2009morphic authors: Yann Bugeaud, Dalia Krieger, and Jeffrey Shallit year: 2009 title: “Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation” doi: null url: https://arxiv.org/abs/0808.2544v2 claim: “Lemmas 10-12 and Corollary 13 give support stabilization and the predecessor/successor mechanism for sufficiently late and long maximal blocks in a morphic word, with bounded boundary corrections.” strata_touched:
- D5/S1/Recurrence/Raney/MaximalBlockEvolution
- D5/S1/Recurrence/Raney/MaximalBlockDescent
- D5/S1/Recurrence/Raney/BoundaryPivotTransport
- D5/S1/Recurrence/Raney/FinitePathDisplacements license: citation-only triage: anchor
Morphic words and maximal blocks
Verified locator
The primary source is arXiv:0808.2544v2 by Yann Bugeaud, Dalia Krieger, and Jeffrey Shallit. Version 1 was submitted in August 2008 and version 2 in April 2009. The source is https://arxiv.org/abs/0808.2544v2. The relevant primary body, not only the abstract or final limsup theorem, was read for this delivery.
Attested mechanism
Lemma 10 replaces a morphism by a positive power for which the set of letters reachable from each letter is stable under every further positive iterate.
Lemma 11 takes a sufficiently late and sufficiently long maximal block over a chosen letter set. Its inverse-scale interval has a central block over that set, while bounded neighborhoods at both ends contain complementary letters.
Lemma 12 runs the construction forward: after trimming bounded boundary regions from a predecessor block, its morphic image remains inside the chosen letter set and bounded image neighborhoods contain complementary letters.
Corollary 13 organizes sufficiently late and long maximal blocks into image evolution sequences with bounded edge corrections. Bugeaud, Krieger, and Shallit use this mechanism to obtain algebraic/rational information about a limsup associated with maximal blocks.
Repository boundary
The support-stabilizing power and the predecessor/successor mechanism are literature prerequisites. The repository specializes them to uniform fixed words and makes every endpoint condition explicit.
The following are repository adaptations and strengthening, not statements attributed to the BKS abstract or its limsup conclusion: the boundary convention at index zero; literal actual maximal intervals; finite descent chains and their roots; finite families of early roots, late root words, and paired edge contexts; deterministic paired pivot states; eventual periodicity of signed left and right endpoint displacements; factorial normalization of those periods; and one finite coefficient set covering every actual maximal block. In particular, the source alone does not assert the repository endpoint that all actual block lengths lie in finitely many affine prime-power families.