Classifying Mixing Smale Spaces by Homoclinic $C^*$-Algebras

Definitions and mathematical statement

A Smale space $(X,\varphi)$ consists of a compact metric space $X$ and a homeomorphism $\varphi\colon X\to X$ satisfying the Smale local product axioms: nearby points have a compatible local stable--unstable product structure, with uniform contraction along local stable sets under forward iteration and along local unstable sets under backward iteration. It is mixing if, for every pair of nonempty open sets $U,V\subseteq X$, there exists $N\in{\mathbb{N}}$ such that \[ \varphi^n(U)\cap V\ne\varnothing\qquad\text{for every integer }n\ge N. \] For such a space, define homoclinic equivalence by \[ x\sim_h y\quad\Longleftrightarrow\quad \lim_{n\to\infty}d(\varphi^n x,\varphi^n y)=0 \quad\text{and}\quad \lim_{n\to\infty}d(\varphi^{-n}x,\varphi^{-n}y)=0. \] Write $H(X,\varphi)$ for the homoclinic $C^*$-algebra: the reduced groupoid $C^*$-algebra of this equivalence relation equipped with its standard Smale-space étale groupoid topology.

For a unital $C^*$-algebra $A$, use the standard Elliott-invariant convention \[ \operatorname{Ell}(A)= \left(K_0(A),K_0(A)^+,[1_A],K_1(A),T(A),r_A\right). \] Here $K_0(A)$ and $K_1(A)$ are its operator $K$-theory groups, $K_0(A)^+$ is the positive cone, $[1_A]$ is the unit class, $T(A)$ is the tracial state space, and $r_A$ is the trace pairing with $K_0(A)$. An isomorphism of these invariants preserves the ordered group, unit, graded group data, trace space, and pairing. The supplied record does not specify a computational procedure or a more particular convention for this invariant.

Source question, with an unresolved target. For mixing Smale spaces, determine whether the computable Elliott invariant of the homoclinic $C^*$-algebra can be used to establish an equivalence of the underlying Smale spaces. A possible converse-classification formulation has the schema \[ \begin{gathered} \text{for all mixing Smale spaces }(X,\varphi),(Y,\psi),\\ \operatorname{Ell}\left(H(X,\varphi)\right) \cong \operatorname{Ell}\left(H(Y,\psi)\right) \quad\Longrightarrow\quad (X,\varphi)\mathrel{\mathcal E}(Y,\psi). \end{gathered} \] The relation $\mathcal E$ is a placeholder for the intended equivalence of Smale spaces; it is not defined by the supplied record. Thus this display is a formulation schema, not a fully specified conjecture attributed to the source. In particular, the record does not justify selecting topological conjugacy or any other particular equivalence. It also does not specify whether additional dynamical data associated with the algebra are intended.

The supplied dated curation asserts that the Elliott invariant classifies the homoclinic algebras in the stated class and labels the dynamical reconstruction question open. Those are supplied assessments, not mathematical or literature claims independently verified in this draft.

Short English statement

The question asks whether the Elliott invariant of a mixing Smale space's homoclinic $C^*$-algebra can recover an equivalence of the underlying dynamical systems. The intended equivalence is unspecified, so a precise converse-classification conjecture remains unresolved at the formulation level. The supplied curation labels the question open.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003207$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2016.
• Supplied report citation, with title typography normalized: $C^*$-Algebras (2017), Oberwolfach Reports, pp. 2269--2345, DOI 10.4171/owr/2016/40. https://doi.org/10.4171/owr/2016/40. The record describes this collection as drawn from mathematical workshop reports and their open-problem discussions.
• Supplied literature source: "OWR Operator Algebras report." https://ems.press/content/serial-article-files/46645?nt=1.
• Supplied literature source: Deeley and Strung, "Group actions on Smale space C star algebras." https://arxiv.org/abs/1709.01897.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and states that the Elliott invariant classifies the homoclinic algebras in the stated class, while recovering an equivalence of the Smale spaces remains open. This is supplied dated curation, not an independently verified assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports that the official report was checked and explicitly poses this converse-classification question after obtaining a computable invariant for the homoclinic algebras. This records the supplied verification claim; no source was visited or independently checked for this draft.