Spectral Bifurcations and Entropy Changes

Definitions and mathematical statement

Formulation and unresolved qualifications. The supplied record asks whether an observed correspondence between bifurcations of the Perron--Frobenius eigenspectrum and changes in topological entropy extends beyond the studied lid-driven-cavity flows. It does not specify the family of systems, the operator space, the spectral events, or the meaning of the correspondence. Consequently, it does not determine a precise universally quantified conjecture.

For clarity, a possible framework for defining the terms is the following; these are conditional definitions, not additional hypotheses supplied by the source. Let $X$ be a compact metric space with metric $d$, let $m$ be a Borel probability measure on $X$, and let $(T_\lambda)_{\lambda\in\Lambda}$ be a family of continuous maps $X\to X$, indexed by a parameter space $\Lambda$. Suppose each map is nonsingular with respect to $m$, meaning that $m(T_\lambda^{-1}A)=0$ whenever $m(A)=0$. Its Perron--Frobenius operator $\mathcal P_\lambda$ on $L^1(m)$ is characterized by \[ \int_X (\mathcal P_\lambda f)g\,{\,\mathrm{d}} m =\int_X f(g\circ T_\lambda)\,{\,\mathrm{d}} m, \qquad f\in L^1(m),\quad g\in L^\infty(m). \] Any use of another function space, an operator approximation, or a time map of a flow must be specified separately. In particular, the relevant eigenvalues and their bifurcations cannot be identified without this choice and a definition of a bifurcation event.

In this conditional compact continuous setting, define \[ d_{\lambda,n}(x,y)=\max_{0\leq j<n}d(T_\lambda^j x,T_\lambda^j y). \] Let $s_\lambda(n,\varepsilon)$ be the largest cardinality of a set whose distinct points have $d_{\lambda,n}$-distance greater than $\varepsilon$. The topological entropy is \[ h_{\mathrm{top}}(T_\lambda) =\lim_{\varepsilon\downarrow0}\limsup_{n\to\infty} \frac{1}{n}\log s_\lambda(n,\varepsilon). \] The supplied record does not establish that this framework is the intended setting. For a flow, the time map and time normalization would also need to be fixed.

Source question. Under what hypotheses, and for what classes of dynamical systems beyond the studied lid-driven-cavity flows, does the reported correspondence between Perron--Frobenius spectral bifurcations and changes in topological entropy persist? A precise assertion requires specifying whether the correspondence is an implication, an equivalence, a quantitative bound, or another relation, and whether an entropy change means a change in value, a transition from zero to positive entropy, a discontinuity, or another event. The original wording refers only to "system entropy"; the supplied cleaned statement specifies topological entropy. No particular interpretation of these unresolved features is asserted here.

Short English statement

The question asks whether the reported connection between Perron--Frobenius spectral bifurcations and topological entropy changes extends beyond the studied lid-driven-cavity flows. The source record leaves the class of systems and the exact connection unspecified, so this remains a qualified research question requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002488$, title Spectral Bifurcations and Entropy Changes, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives the proposed year as 2014 and supplies no individual proposer.
• Supplied report citation: Mixing, Transport and Coherent Structures (2014), Oberwolfach Reports, pp. 213--286, DOI 10.4171/owr/2014/04. https://doi.org/10.4171/owr/2014/04. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions; it does not identify a specific page or contribution for this question.
• Supplied literature source, labelled "almost-cyclic-set bifurcation study": https://doi.org/10.1063/1.4768666. The record supplies no authors, full title, or pinpoint result for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the reported correspondence is proved only in model settings, that the primary follow-up develops the almost-cyclic-set method and model-specific entropy bounds rather than a general theorem, and that the proposed generalization remains open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report, with the cleaned statement restoring source hypotheses and notation. These are supplied verification claims; neither source access nor independent verification is claimed in this draft.