Flow equivalence of subshifts

Definitions and mathematical statement

Let $(X,\sigma)$ be a nonempty two-sided subshift over a finite alphabet. Its unit suspension is the compact space \[ SX=(X\times\mathbb R)/\!\sim,\qquad (x,t+1)\sim(\sigma x,t), \] with suspension flow $\phi_X^s[x,t]=[x,t+s]$. Two subshifts $X,Y$ are flow equivalent if there is a homeomorphism $H:SX\to SY$ carrying every flow orbit onto a flow orbit and preserving its time orientation. Equivalently, along each orbit $H$ admits an increasing continuous reparametrization of time; preservation of the actual elapsed time is not required.

Classify subshifts under this relation: give complete invariants, or a usable necessary and sufficient criterion, for $X$ and $Y$ to be flow equivalent. The question includes arbitrary finite-alphabet subshifts, not only those with finite forbidden-word lists or finite labelled-graph presentations.

The homeomorphism must be defined on the entire compact suspension, including closed flow orbits coming from periodic shift configurations and all their accumulation points. On a closed orbit it preserves the circle's orientation; on a nonclosed orbit it preserves the forward direction. It need not send the chosen cross-section $X\times\{0\}$ to $Y\times\{0\}$. This permits changes in return times which ordinary topological conjugacy forbids. The problem asks for a classification of this precise orbit-and-orientation relation, without imposing a finite input encoding for all subshifts.

Short English statement

Classify finite-alphabet subshifts when their suspension flows are identified by orientation-preserving orbit homeomorphisms.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4600052 (AMR-045-0052), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Boyle - Open Problems in Symbolic Dynamics (2008), Problem/Question/Conjecture 33.3. Original source indexed by AMR-045-0052. https://www.math.umd.edu/~mboyle/papers/openfinalsub3nov2007.pdf.