Borel Structure and Generic Triviality of Flow Self-Similarities

Definitions and mathematical statement

In the standard probability-space interpretation, let $T=(T_t)_{t\in{\mathbb{R}}}$ be a measurable measure-preserving flow on a standard nonatomic probability space $(X,\mathcal B,\mu)$. Thus $T_0$ is the identity and $T_{t+u}=T_tT_u$ as measure-preserving transformations modulo null sets. The source does not explicitly specify this space class, finiteness of measure, or ergodicity.

For $s\in{\mathbb{R}}^\times={\mathbb{R}}\setminus\{0\}$, write $T^{(s)}_t=T_{st}$. Define the self-similarity group, in the nonzero-real convention, by \[ I(T)=\{s\in{\mathbb{R}}^\times:\exists R\in\operatorname{Aut}(X,\mu) \text{ with }RT_tR^{-1}=T_{st}\text{ for every }t\in{\mathbb{R}}\}. \] The same measure-space isomorphism $R$ must implement all times. Composition and inverse conjugacies show that $I(T)$ is a multiplicative subgroup and contains $1$. An alternative convention restricts to positive $s$; then $I_+(T)=I(T)\cap{\mathbb{R}}_{>0}$. The value $s=-1$ concerns time reversal and makes these conventions materially different.

The first question asks whether $I(T)$ is always a Borel subset of ${\mathbb{R}}^\times$ with its usual topology, hence a Borel subgroup with the inherited operations. If "Borel group" instead means an abstract standard-Borel or Polishable group structure, that is a different specification not provided by the record.

The second question asks whether absence of nontrivial self-similarity is generic. Under the nonzero-real convention this means the class \[ \{T:I(T)=\{1\}\} \] is comeagre in the intended space of measure-preserving flows; under the positive convention replace $I(T)$ by $I_+(T)$. A usual topology is uniform convergence on compact time intervals for the weak topology on $\operatorname{Aut}(X,\mu)$. The source does not fix that topology or whether genericity is in all flows, ergodic flows, or another subclass. Genericity is not interpreted as a probability-one statement without a specified probability measure.

The supplied assessment dated 2026-08-21 says recent work continues to list the Borel and related genericity questions as open. The supplied secondary reference concerns ergodic flows, while the extracted question says any measurable measure-preserving flow; this scope difference requires review.

Short English statement

Is the set of time-rescaling factors producing an isomorphic measure-preserving flow always Borel, and do generic flows have only the trivial rescaling? The sign convention, underlying measure space, ergodicity scope, and topology for genericity must be fixed.

Sources

• Local UnsolvedMath registry, permanent ID $30001032$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406. https://doi.org/10.4171/owr/2008/42.
• Supplied reference labelled "Self-similarity problem for ergodic flows": https://www-users.mat.umk.pl/~fraczek/czasypot2.pdf.
• The supplied corrected_verified note reports only notation cleanup. The open label and assessment checked 2026-08-21 are supplied dated curation, not independently verified; the exact definitions, genericity setting, and current status require review.