An ergodic classification of smooth type subshifts over odd two-letter alphabets

Will Howe; Grant Huffman; Jensen Kohlmeyer; Liam Kruer

An ergodic classification of smooth type subshifts over odd two-letter alphabets

Primarily AI-generated textHuman understanding: all partsmath.DS — Dynamical Systems

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer ↗, Liam Kruer

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Fix distinct positive odd integers

\alpha<\beta

. We classify the invariant probability measures of the smooth subshifts obtained by prescribing the successive run types. Every such subshift is uniquely ergodic if

\alpha\ge3

or

\{\alpha,\beta\}=\{1,3\},\{1,5\}

. For

\alpha=1

and

\beta\ge7

, unique ergodicity holds exactly for directives containing infinitely many zeros; every other directive gives precisely two ergodic measures. The proof combines a two-dimensional frequency recursion with recognized run decompositions and explicit control of the density of decomposition boundaries.

Provenance statement

The mathematical draft and exposition were produced primarily with OpenAI Codex under the direction of the listed contributors. Lean was used to verify the finite frequency-map calculations and the compatible-frequency-chain classification.

Tools used

OpenAI: Codex
Lean: Lean

References

D. Jamet, I. Marcovici, L. Poirier, and T. de la Rue, Frequency of patterns in smooth sequences over the alphabet $\{1,3\}$ , arXiv:2604.11387v3 [math.DS], 2026. https://arxiv.org/abs/2604.11387v3.arXiv

Version history

v1 · 9 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 11, 2026