Ebin Analysis for Anosov Diffeomorphism Moduli

Definitions and mathematical statement

A conventional framework for interpreting the supplied questions is the following; the source record does not specify that this is the intended full set of hypotheses. Let $M$ be a closed smooth manifold, let $\operatorname{Diff}^{\infty}(M)$ denote its group of smooth diffeomorphisms, and fix an auxiliary Riemannian norm on $TM$. A diffeomorphism $f\in\operatorname{Diff}^{\infty}(M)$ is Anosov if there exist a continuous $Df$-invariant splitting $TM=E^s\oplus E^u$, a constant $C\geq 1$, and $0<\lambda<1$ such that, for every $x\in M$, every integer $n\geq0$, and all $v^s\in E_x^s$, $v^u\in E_x^u$, \[ \|D(f^n)_xv^s\|\leq C\lambda^n\|v^s\|, \qquad \|D(f^{-n})_xv^u\|\leq C\lambda^n\|v^u\|. \] Write $\mathcal A^{\infty}(M)$ for the set of such diffeomorphisms. Two elements $f,g\in\mathcal A^{\infty}(M)$ are smoothly conjugate if there is $h\in\operatorname{Diff}^{\infty}(M)$ with $g=h\circ f\circ h^{-1}$. Define the smooth-conjugacy class and the corresponding set of equivalence classes by \[ \mathcal O^{\infty}(f)=\{h\circ f\circ h^{-1}:h\in\operatorname{Diff}^{\infty}(M)\}, \qquad \mathcal M_A^{\infty}(M)=\mathcal A^{\infty}(M)/{\sim}, \] where $\sim$ is smooth conjugacy.

For a metric space $(X,d)$ and $S\subseteq X$, define, for $s>0$, \[ \mathcal H_d^s(S)=\lim_{\delta\downarrow0}\inf\left\{ \sum_{i=1}^{\infty}(\operatorname{diam}_d U_i)^s: S\subseteq\bigcup_{i=1}^{\infty}U_i,\quad \operatorname{diam}_d U_i\leq\delta \right\}, \qquad \dim_{H,d}S=\inf\{s>0:\mathcal H_d^s(S)=0\}, \] with $\inf\varnothing=+\infty$.

Questions recorded by the source. Can a smooth-conjugacy class of Anosov diffeomorphisms have Hausdorff dimension greater than $2$? Can Ebin's infinite-dimensional analysis of Riemannian metrics modulo diffeomorphisms be adapted to Anosov diffeomorphisms and their moduli spaces?

Unresolved formulation. Under the provisional framework above, the first question would ask whether $\dim_{H,d}\mathcal O^{\infty}(f)>2$ can occur for an admissible $M$ and $f$, but the record supplies neither the metric $d$ nor the admissible manifolds, their dimensions, or the intended differentiability classes. Hausdorff dimension depends on the metric; a topology alone does not determine it. Thus this expression does not yet specify a unique mathematical problem. The second question is a research-program question: the record does not identify the particular Ebin results or analytical structures to be reproduced, the topology or geometry of the moduli space, or the criterion for a successful adaptation. The definitions above provide terminology and do not resolve these omissions. The original statement contains only the second question, whereas the supplied rewritten statements also contain the Hausdorff-dimension question.

Short English statement

The supplied record asks whether a smooth-conjugacy class of Anosov diffeomorphisms can have Hausdorff dimension greater than $2$, and whether Ebin's methods for Riemannian metrics modulo diffeomorphisms extend to this setting. A precise statement requires the intended metric, underlying manifolds, regularity, and analytical goals. The record labels the problem open; that label is supplied curation and has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003367$, title Ebin Analysis for Anosov Diffeomorphism Moduli, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2017$ and names no proposer.
• Supplied source citation: Mini-Workshop: Spaces and Moduli Spaces of Riemannian Metrics (2018), Oberwolfach Reports, pp. 133--166, DOI 10.4171/owr/2017/3. https://doi.org/10.4171/owr/2017/3. The citation's parenthetical year $2018$ differs from the record's proposed year and its identification as Report 3/2017; this discrepancy remains unresolved.
• Supplied literature source: Oberwolfach Report 3/2017. https://ems.press/content/serial-article-files/46667.
• Supplied literature source: Smooth conjugacy of Anosov diffeomorphisms. https://arxiv.org/abs/0804.3901.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that no general Ebin-style infinite-dimensional analysis for the Anosov setting is known, while existing work parametrizes special smooth-conjugacy classes and establishes rigidity only in restricted families. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry assigns statement_status: corrected_verified and reports that the official passage was checked and rewritten while preserving its mathematical content. These are supplied provenance claims; no independent source inspection is asserted here. Ebin is named in the problem, but no specific bibliographic reference to his analysis is supplied.